Matrices

01

PYQ 2023

medium

mathematics ID: up-board

Solve the following system of equations by matrix method:

02

PYQ 2023

medium

mathematics ID: up-board

Find the inverse of the matrix by elementary transformations.

03

PYQ 2023

medium

mathematics ID: up-board

If prove that

04

PYQ 2023

medium

mathematics ID: up-board

Prove:

05

PYQ 2023

easy

mathematics ID: up-board

If find the matrix .

06

PYQ 2023

medium

mathematics ID: up-board

If and, then will be:

1

2

3

4

07

PYQ 2023

medium

mathematics ID: up-board

Define symmetric and skew-symmetric matrices. If and are symmetric matrices, prove that is a skew-symmetric matrix.

08

PYQ 2023

medium

mathematics ID: up-board

Solve by matrix method the system of equations:

09

PYQ 2023

medium

mathematics ID: up-board

$Prove that the matrix satisfies the equation where is the identity matrix and is the zero matrix. Also, find with the help of this.$

Official Solution

Correct Option: (1)

$Step 1: Compute . Step 2: Compute . Hence proved. ✅ Step 3: Find . From Cayley-Hamilton theorem: Final Answer:$

10

PYQ 2023

medium

mathematics ID: up-board

$Solve the system of equations by matrix method:$

Official Solution

Correct Option: (1)

$Step 1: Write system in matrix form. where Step 2: Solve using inverse method. Step 3: Find . So,, hence solution exists. Step 4: Use Cramer's Rule. - Replace first column of with for : - Replace second column of with for : - Replace third column of with for : Step 5: Compute values. Final Answer:$

11

PYQ 2023

medium

mathematics ID: up-board

$If and, then prove that .$

Official Solution

Correct Option: (1)

$Step 1: Compute . Step 2: Find .} For a matrix, Here, . Step 3: Find and .} For, . For, . Step 4: Compute . Step 5: Compare results. Final Answer:$

12

PYQ 2023

medium

mathematics ID: up-board

$If and are two non-singular square matrices of order, then prove that$

Official Solution

Correct Option: (1)

$Step 1: Recall definition. For any non-singular matrix, Step 2: Take . Step 3: Conclude. Since, it follows that Final Answer:$

13

PYQ 2023

medium

mathematics ID: up-board

$If then the value of and will be:$

1

2

3

4

Official Solution

Correct Option: (1)

$We are given the matrix equation: Step 1: Compare corresponding elements of the matrices. From the equation, we can equate the corresponding elements of both matrices. 1. From the top-left element: 2. From the top-right element: 3. From the bottom-left element: 4. From the bottom-right element: Step 2: Solve the system of equations. We now have the system of equations: 1. 2. Let's solve this system. From Equation 1, solve for : Substitute this into Equation 2: Now substitute into the expression for : Step 3: Conclusion. The solution to the system is and, so the correct answer is (A) .$

14

PYQ 2023

medium

mathematics ID: up-board

$Let Then find a matrix D such that .$

Official Solution

Correct Option: (1)

$We are asked to find a matrix such that: This can be rewritten as: We need to calculate and then solve for . Step 1: Calculate . First, compute the product : Using matrix multiplication: Step 2: Solve for . Now, we know, so we have the equation: Let . Now, multiply with : Equating this with, we get the system of equations: Step 3: Solve the system of equations. From the first equation, solve for : Substitute into the second equation: Simplifying: Now substitute into the equation for : Now solve for and : From : Substitute this into the second equation: Simplifying: Substitute into the equation for : Thus, the matrix is: Final Answer: The matrix such that is:$

15

PYQ 2023

medium

mathematics ID: up-board

$Solve the system of linear equations by matrix method:$

Official Solution

Correct Option: (1)

$We can solve the given system of linear equations using the matrix method. The system of equations can be written in matrix form as: Let be the coefficient matrix, be the variable matrix, and be the constant matrix: Now, solve for using the inverse of matrix : Step 1: Find the determinant of . \\ The determinant of , denoted as , is calculated as: Step 2: Find the inverse of . To find , we use the formula: The adjugate matrix is the transpose of the cofactor matrix of , which is computed as follows. Step 3: Multiply with . Finally, multiply the inverse of by to find . This will give the values of , , and . Conclusion: The solution for the system of equations will provide the values of , , and .$

16

PYQ 2023

medium

mathematics ID: up-board

$If and$

Official Solution

Correct Option: (1)

$We are given the matrix equation: Step 1: Isolate . First, subtract from both sides to isolate : Now, multiply both sides by to solve for : Step 2: Calculate . First, calculate and : Now subtract from : Step 3: Find . Now, divide each element of the matrix by 3: Conclusion: The matrix is:$

17

PYQ 2023

medium

mathematics ID: up-board

$If verify that .$

Official Solution

Correct Option: (1)

$Step 1: Write, the transpose of . Step 2: Multiply . Final Answer:$

18

PYQ 2023

medium

mathematics ID: up-board

$If is a square matrix such that, then will be:$

1

2

3

4

Official Solution

Correct Option: (3)

$Step 1: Expand .} Step 2: Use the condition .} Since, So, Step 3: Simplify expression. Step 4: Conclusion. The correct answer is (C) .$

19

PYQ 2024

medium

mathematics ID: up-board

$Find the values of if the matrix satisfies the equation, where$

Official Solution

Correct Option: (1)

$Step 1: Compute (transpose of). Step 2: Compute . Step 3: Solve for using . Comparing with identity matrix:$

20

PYQ 2024

easy

mathematics ID: up-board

$(a) If orders of matrices A and B are and respectively, then order of AB is:$

1

2

3

4

$None of these$

Official Solution

Correct Option: (1)

$Understanding matrix multiplication order. If is of order and is of order, then the resulting matrix has order:$

21

PYQ 2024

easy

mathematics ID: up-board

$Solve the system of equations by matrix method:$

Official Solution

Correct Option: (1)

22

PYQ 2024

easy

mathematics ID: up-board

$If matrix then find .$

Official Solution

Correct Option: (1)

23

PYQ 2024

easy

mathematics ID: up-board

$If prove that where .$

Official Solution

Correct Option: (1)

24

PYQ 2024

easy

mathematics ID: up-board

$Find matrix if$

Official Solution

Correct Option: (1)

25

PYQ 2024

medium

mathematics ID: up-board

$Solve by matrix method the system of equations:$

Official Solution

Correct Option: (1)

$Step 1: Write the system of equations in matrix form: Step 2: Find the inverse of the coefficient matrix: Step 3: Use the formula to solve for, where represents the vector of variables and is the constant vector. After calculating and multiplying by, the solution is: Thus, the solution to the system of equations is .$

26

PYQ 2024

medium

mathematics ID: up-board

$If then verify that and find .$

Official Solution

Correct Option: (1)

$Step 1: To verify that, we first calculate the determinant of matrix . The determinant is given by: After performing the calculation, . Step 2: The adjugate of, denoted, is the transpose of the cofactor matrix of . Calculate the cofactor matrix and then its transpose to find . Step 3: Verify the identity, where is the identity matrix. Step 4: To find, use the formula: Substitute and the calculated adjugate to find . Thus, the inverse of is:$

27

PYQ 2025

hard

mathematics ID: up-board

$If, find . Using solve the following system of equations: 2x - 3y + 5z = 11 3x + 2y - 4z = -5 x + y - 2z = -3.$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This problem has two parts. First, we need to find the inverse of a 3x3 matrix. Second, we use this inverse to solve a system of linear equations. The system of equations is given in the form AX = B, where A is the same matrix for which we need to find the inverse. The solution is given by X = A B. Step 2: Key Formula or Approach: Part 1: Finding 1. Calculate the determinant of A, . 2. Find the matrix of cofactors, C. 3. Find the adjugate of A, adj(A), which is the transpose of C. 4. The inverse is . Part 2: Solving the system AX = B 1. Identify the matrices A, X, and B from the system of equations. 2. Calculate the solution using X = A B. Step 3: Detailed Explanation: Part 1: Finding Given . 1. Determinant of A: Since, the inverse exists. 2. Adjugate of A: The cofactor matrix C is: So, . adj(A) = . 3. Inverse of A: Part 2: Solving the System of Equations The system is: This can be written as AX = B, where: The solution is X = A B. Step 4: Final Answer: The inverse is . The solution to the system of equations is x = 1, y = 2, z = 3.$

28

PYQ 2025

easy

mathematics ID: up-board

$For two invertible matrices A and B of order n, prove that .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: We need to prove the reversal law for the inverse of a product of matrices. The proof relies on the fundamental definition of a matrix inverse: a matrix M is the inverse of N if their product, in either order, is the identity matrix I (i.e.,). Step 2: Key Formula or Approach: To prove that is the inverse of, we need to show that: 1. 2. Step 3: Detailed Explanation or Calculation: Let's start with the product and . Proof of the first condition: Using the associative property of matrix multiplication, we can regroup the terms: Since B is an invertible matrix, by definition, (the identity matrix). The product of any matrix with the identity matrix is the matrix itself (). Since A is an invertible matrix, by definition, . So, we have shown that . Proof of the second condition: Now, let's check the product in the reverse order: Using the associative property: Since A is invertible, . Since : Since B is invertible, . So, we have also shown that . Step 4: Final Answer: Since and, by the definition of an inverse, the inverse of the matrix is . Hence, is proved.$

29

PYQ 2025

medium

mathematics ID: up-board

$If, then show that . Using this, obtain .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: We use the Cayley-Hamilton theorem: a square matrix satisfies its own characteristic equation. First, verify the equation . Then, manipulate it to find . Step 2: Detailed Explanation: Part 1: Verify Calculate : Calculate : Calculate : Substitute: Part 2: Find Start from . Multiply by : Solve for : Substitute values: Step 3: Final Answer:$

30

PYQ 2025

medium

mathematics ID: up-board

$If, then find .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: To find the inverse of a square matrix, we use the formula, where is the determinant of and is the adjugate (or adjoint) of . The inverse exists only if the determinant is non-zero. Step 2: Key Formula or Approach: The process involves three main steps: 1. Calculate the determinant of, . 2. Find the matrix of cofactors, then transpose it to get the adjugate matrix, . 3. Substitute these values into the inverse formula. Step 3: Detailed Explanation: 1. Calculate the Determinant of A: Since, the inverse exists. 2. Find the Adjugate of A: The adjugate is the transpose of the cofactor matrix. Cofactors are: Cofactor matrix: Adjugate matrix: 3. Calculate the Inverse: Step 4: Final Answer: The inverse of the matrix is$

31

PYQ 2025

medium

mathematics ID: up-board

$If and, then prove that, where .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This problem involves matrix operations, including transpose, multiplication, and finding the determinant. The key is to first simplify the expression for matrix and then perform the necessary calculations to find its determinant. Step 2: Key Formula or Approach: 1. Simplify the expression for using the matrix property . 2. Perform the matrix multiplication to find the elements of . 3. Calculate the determinant of the resulting matrix using the formula . Step 3: Detailed Explanation: We are given the definition of as: Using the property that the transpose of a transpose of a matrix is the matrix itself, i.e.,, the expression for simplifies to: Now we multiply matrix by matrix : The resulting matrix will be . Calculating its elements: So, the matrix is: Finally, calculate the determinant of : Step 4: Final Answer: The determinant of is . Hence proved.$

32

PYQ 2025

medium

mathematics ID: up-board

$If and are two matrices of order which are invertible, then prove that .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: The inverse of a square matrix M is a matrix such that their product is the identity matrix I, i.e., . To prove the given identity, we must show that multiplying by results in the identity matrix. Step 2: Key Approach: We will use the definition of an inverse. We will show that . Step 3: Detailed Proof: Consider the product of the matrix and the matrix : Using the associative property of matrix multiplication, we can regroup the terms: Since B is an invertible matrix, by definition,, where I is the identity matrix of order n. The product of any matrix and the identity matrix is the matrix itself, so . Since A is an invertible matrix, by definition, . Thus, we have shown that . This proves that is the right inverse of . For square matrices, a right inverse is also a left inverse, so is the inverse of . Step 4: Final Answer: By definition of a matrix inverse and the associative property of multiplication, we have proved that .$

33

PYQ 2025

easy

mathematics ID: up-board

$If, then determine a matrix A of order .$

Official Solution

Correct Option: (1)

34

PYQ 2025

medium

mathematics ID: up-board

$Solve by matrix method the following system of equations:$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: The matrix method for solving a system of linear equations involves representing the system in the form . The solution is then found by calculating, where is the inverse of the coefficient matrix . This method is applicable only if the determinant of is non-zero. Step 2: Key Formula or Approach: 1. Write the system of equations in the matrix form . 2. Calculate the determinant of A, . 3. Find the adjugate of A, . 4. Calculate the inverse of A using the formula . 5. Find the solution by computing . Step 3: Detailed Explanation: The given system of equations can be written as: In matrix form, we have: First, calculate the determinant of A: Since, a unique solution exists. Next, find the adjugate of A. The matrix of cofactors is: The matrix of cofactors is . The adjugate of A is the transpose of the cofactor matrix: Now, find the inverse of A: Finally, calculate the solution : Step 4: Final Answer: The solution to the system of equations is,, and .$

35

PYQ 2025

medium

mathematics ID: up-board

$For matrix show that and with the help of this find .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This problem has two parts. First, we need to verify that the matrix satisfies the given polynomial equation, which is its characteristic equation (Cayley-Hamilton Theorem). Second, we use this verified equation to find the inverse of matrix without using the adjugate method. Step 2: Verifying the Matrix Equation : First, we calculate and . Now, substitute and into the equation: Thus, the equation is verified. Step 3: Finding using the equation: Starting with the verified equation: Multiply the entire equation by : Isolating : Substitute the matrices: Step 4: Final Answer: The matrix equation is verified, and the inverse is:$

36

PYQ 2025

easy

mathematics ID: up-board

$If the matrices and, then prove that .$

Official Solution

Correct Option: (1)

37

PYQ 2025

easy

mathematics ID: up-board

$If, prove that .$

Official Solution

Correct Option: (1)

38

PYQ 2025

easy

mathematics ID: up-board

$If and is the identity matrix of order 2, prove that .$

Official Solution

Correct Option: (1)

39

PYQ 2025

easy

mathematics ID: up-board

$If and, then will be$

Official Solution

Correct Option: (1)

40

PYQ 2025

easy

mathematics ID: up-board

$If and, find .$

Official Solution

Correct Option: (1)

41

PYQ 2025

easy

mathematics ID: up-board

$If, verify that .$

Official Solution

Correct Option: (1)

42

PYQ 2025

hard

mathematics ID: up-board

$If \( A = \begin{bmatrix} 0 & -\tan{\frac{\alpha}{2}} \$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This problem requires proving a matrix equality. The strategy is to compute the left-hand side (LHS) and the right-hand side (RHS) of the equation separately and then show that they are equal. The proof will heavily rely on trigonometric half-angle identities. Step 2: Key Formula or Approach: We will use the half-angle identities for sine and cosine in terms of tangent: Let . Then and . Step 3: Detailed Explanation or Calculation: Let . The matrix is Calculate the LHS: Calculate the RHS: First, find : The rotation matrix is Now compute the product : Now substitute the half-angle formulas for and : Entry (1,1): Entry (1,2): Entry (2,1): Entry (2,2): So, the RHS matrix is: Step 4: Final Answer: We have shown that LHS and RHS . Since LHS = RHS, the identity is proved.$

43

PYQ 2025

medium

mathematics ID: up-board

$Find the inverse of the matrix .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: To find the inverse of a matrix A, we use the formula, where is the determinant of A and is the adjugate (or adjoint) of A. The inverse exists only if the determinant is non-zero. Step 2: Key Formula or Approach: 1. Calculate the determinant of A, . 2. Find the matrix of cofactors of A. 3. Find the adjugate of A by transposing the cofactor matrix, . 4. Calculate the inverse using the formula . Step 3: Detailed Explanation or Calculation: 1. Calculate the Determinant: Expanding along the first row: Since, the inverse exists. 2. Find the Cofactor Matrix: The cofactors, where is the minor of the element . The cofactor matrix is . 3. Find the Adjugate Matrix: 4. Calculate the Inverse: Step 4: Final Answer: The inverse of the matrix A is:$

44

PYQ 2025

medium

mathematics ID: up-board

$If the orders of the matrices A and B are and respectively, then the order of AB will be$

1

2

3

4

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This question tests the fundamental rule for the multiplication of two matrices. The product of two matrices, AB, is defined only if the number of columns in the first matrix (A) is equal to the number of rows in the second matrix (B). The resulting matrix has an order determined by the outer dimensions of the original matrices. Step 2: Key Formula or Approach: Let matrix A have order . Let matrix B have order . The condition for multiplication is that the inner dimensions must match: . This is satisfied. The order of the product matrix AB is given by the outer dimensions: . Step 3: Detailed Explanation: We are given: Order of matrix A = (m rows, n columns). Order of matrix B = (n rows, p columns). To find the order of the product AB, we look at the dimensions: The number of columns of A is, and the number of rows of B is . Since they are equal, the product AB is defined. The order of the resulting matrix AB is the number of rows of A by the number of columns of B. Therefore, the order of AB is . Step 4: Final Answer: The order of the product matrix AB is .$

45

PYQ 2025

medium

mathematics ID: up-board

$Solve the following system of equations by matrix method : 2x + 3y + 3z = 5 x - 2y + z = -4 3x - y - 2z = 3$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: A system of linear equations can be represented in matrix form as . The solution can be found using the formula, provided that the coefficient matrix A is invertible (i.e., its determinant is non-zero). Step 2: Setting up the Matrix Equation: The given system of equations can be written as, where: Step 3: Finding the Inverse of the Coefficient Matrix A: First, we calculate the determinant of A to check for invertibility. Since, the inverse exists. Next, we find the adjugate of A. The cofactor matrix C is: So, the cofactor matrix is . The adjugate is the transpose of the cofactor matrix: . The inverse is : Step 4: Solving for X: Now we calculate : Step 5: Final Answer: By comparing the elements of the resulting matrix, we find the solution: and .$

46

PYQ 2025

medium

mathematics ID: up-board

$Solve the following system of equations by matrix method:$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: A system of linear equations can be represented in matrix form as, where A is the matrix of coefficients, X is the column matrix of variables, and B is the column matrix of constants. The solution is found by pre-multiplying both sides by the inverse of A, which gives . Step 2: Key Formula or Approach: 1. Write the system in the form . 2. Calculate the inverse of the coefficient matrix, . 3. Multiply by B to find the solution matrix X. Step 3: Detailed Explanation: 1. Represent the system in matrix form: The equation is, and the solution is . 2. Find the inverse of A: First, calculate the determinant of A: Since, a unique solution exists. Next, find the adjugate of A. The cofactors are: The adjugate matrix is . The inverse is . 3. Solve for X: So, . Step 4: Final Answer: The solution to the system of equations is x = 2, y = 1, and z = -1.$

47

PYQ 2025

medium

mathematics ID: up-board

$If the matrix A = satisfies the equation AA' = I, then find the values of x, y and z.$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: The given equation is, where A' is the transpose of matrix A and I is the identity matrix. A matrix A that satisfies this condition is called an orthogonal matrix. For an orthogonal matrix, the dot product of any row vector with itself is 1, and the dot product of any two distinct row vectors is 0. Step 2: Key Formula or Approach: 1. Find the transpose of matrix A, denoted as A'. 2. Calculate the product AA'. 3. Equate the resulting matrix with the identity matrix . 4. Solve the system of equations obtained by equating the corresponding elements. Step 3: Detailed Explanation or Calculation: Given matrix . The transpose of A is . Now, we compute the product : Equating with : From the non-diagonal elements, we get: (Equation 1) From the diagonal elements, we get: (Equation 2) (Equation 3) Substitute from Eq. 1 into Eq. 2: . Now find using Eq. 1: . Finally, substitute the values of and into Eq. 3 to find x: . . Step 4: Final Answer The values are,, and .$

48

PYQ 2025

medium

mathematics ID: up-board

$If A is a square matrix and, then will be:$

1

$A$

2

$3A$

3

$I - A$

4

$I$

Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: We need to simplify a matrix expression given a specific property of the matrix A (, which means A is an idempotent matrix). We use the properties of matrix algebra, including the binomial expansion for matrices that commute (A and I commute). Step 2: Key Formula or Approach: - Binomial expansion: . - Identity matrix properties: for any positive integer n, and . - Given property: . This also implies . Step 3: Detailed Explanation or Calculation: First, expand : Using the identity matrix properties, this simplifies to: Now, use the given condition . This implies . Substitute these into the expansion: Now, substitute this result back into the original expression: Step 4: Final Answer: The expression simplifies to the identity matrix, I.$

49

PYQ 2025

medium

mathematics ID: up-board

$Solve the following system of linear equations by matrix method:$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: We can solve a system of linear equations by representing it in the matrix form, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. The solution is then found using the formula . Step 2: Key Formula or Approach: 1. Rewrite the equations in standard form and express the system as . 2. Find the inverse of the coefficient matrix A. 3. Multiply by B to find the solution matrix X. Step 3: Detailed Explanation or Calculation: 1. Formulate the matrix equation: First, rewrite the third equation in standard form: . The system is: In matrix form : 2. Find the inverse of A: First, find the determinant of A: Next, find the adjugate of A. The cofactor matrix is: The inverse is: 3. Solve for X: Step 4: Final Answer: The solution is .$

50

PYQ 2025

medium

mathematics ID: up-board

$If and, then find the value of .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: We are given a matrix A and a condition involving the matrix, its transpose (A'), and the identity matrix (I). We need to solve the resulting matrix equation to find the value of the angle . Step 2: Key Formula or Approach: 1. Find the transpose of A, denoted by A', by interchanging its rows and columns. 2. Add the matrices A and A'. 3. Set the resulting matrix equal to the 2x2 identity matrix, . 4. Equate the corresponding elements to form an equation and solve for . Step 3: Detailed Explanation or Calculation: The given matrix is: Its transpose is: Now, add A and A': We are given that : Equating the corresponding elements (specifically, the diagonal elements): The principal value of for which is . Step 4: Final Answer: The value of is .$

51

PYQ 2025

medium

mathematics ID: up-board

$If, then verify that and find .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This question involves two key properties of matrices. First, the relationship between a square matrix, its adjugate (adj A), its determinant (|A|), and the identity matrix (I), which is given by A(adj A) = (adj A)A = |A| I. Second, using this property to find the inverse of the matrix A, given by the formula A -1 = (1/|A|) (adj A). Step 2: Key Formula or Approach: 1. Calculate the determinant of A, |A|. 2. Find the matrix of cofactors of A. 3. Find the adjugate of A, adj(A), which is the transpose of the cofactor matrix. 4. Calculate the product A(adj A). 5. Calculate the product |A|I and verify it is equal to A(adj A). 6. Calculate the inverse A -1 using the formula A -1 = (1/|A|) adj(A). Step 3: Detailed Explanation or Calculation: 1. Calculate the Determinant |A|: |A| = 3(4\cdot1 - 1\cdot3) - 3(3\cdot1 - 1\cdot4) + 1(3\cdot3 - 4\cdot4) |A| = 3(1) - 3(-1) + 1(-7) = 3 + 3 - 7 = -1 Since |A| \neq 0, the inverse of A exists. 2. Find the Adjugate, adj(A): Cofactor matrix C = [[1, 1, -7], [0, -1, 3], [-1, 0, 3]] Adjugate is transpose of C: adj(A) = [[1, 0, -1], [1, -1, 0], [-7, 3, 3]] 3. Verify A(adj A) = |A|I: A(adj A) = [[3,3,1],[3,4,1],[4,3,1]] \cdot [[1,0,-1],[1,-1,0],[-7,3,3]] = [[-1,0,0],[0,-1,0],[0,0,-1]] = |A| I 4. Find A -1 : A -1 = (1/|A|) adj(A) = (-1)\cdot[[1,0,-1],[1,-1,0],[-7,3,3]] A -1 = [[-1,0,1],[-1,1,0],[7,-3,-3]] Step 4: Final Answer: The property A(adj A) = |A|I is verified. The inverse of the matrix A is A -1 = [[-1,0,1],[-1,1,0],[7,-3,-3]].$

52

PYQ 2025

medium

mathematics ID: up-board

$If, then verify that and find .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This problem involves two parts. First, we need to verify a fundamental property of matrices which states that the product of a matrix and its adjugate is equal to the determinant of the matrix times the identity matrix. Second, we need to find the inverse of the matrix A using the formula involving the determinant and the adjugate. Step 2: Key Formula or Approach: 1. Calculate the determinant of A, denoted as . 2. Find the matrix of cofactors, then find the adjugate of A,, which is the transpose of the cofactor matrix. 3. Calculate the product . 4. Calculate the product, where I is the 3x3 identity matrix. 5. Compare the results from steps 3 and 4 to verify the property. 6. Use the formula to find the inverse. Step 3: Detailed Explanation or Calculation: Given . 1. Calculate : 2. Find : The cofactors are: The cofactor matrix is . The adjugate matrix is . 3. Calculate : 4. Calculate : 5. Verification: From steps 3 and 4, we see that is verified. 6. Find : Step 4: Final Answer: The property is verified. The inverse of A is:$

53

PYQ 2025

medium

mathematics ID: up-board

$Express the matrix as the sum of a symmetric and a skew-symmetric matrix.$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: Any square matrix A can be uniquely expressed as the sum of a symmetric matrix P and a skew-symmetric matrix Q. Step 2: Key Formula or Approach: The symmetric part is given by . The skew-symmetric part is given by . Then, . Step 3: Detailed Explanation or Calculation: The given matrix is . First, find the transpose of A: Calculate the symmetric matrix P: Calculate the skew-symmetric matrix Q: Step 4: Final Answer: The matrix A can be expressed as the sum, where:$

54

PYQ 2025

medium

mathematics ID: up-board

$If, then find the values of .$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This problem is based on the principle of equality of matrices. Two matrices are equal if and only if they have the same dimensions and their corresponding elements are equal. Step 2: Key Formula or Approach: Given the matrix equation, where and, the equality implies: - - - - We will set up a system of equations based on this principle and solve for the variables,, and . Step 3: Detailed Explanation or Calculation: The given matrix equation is: By equating the corresponding elements, we get the following system of equations: 1. 2. (This is consistent and provides no new information) 3. 4. Solve for z: From equation (3): Solve for x and y: We have a system of two equations with two variables: (1) (4) From equation (1), express in terms of : Substitute this expression for into equation (4): Rearrange into a standard quadratic equation form : Factor the quadratic equation: This gives two possible values for : or . Now find the corresponding values for using : Case 1: If Case 2: If So, we have two possible sets of solutions for . Step 4: Final Answer: The values of the variables are: - - Either or . The complete solutions are and .$

High-Yield Trend

Chapter Questions 54 MCQs

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Step 1: Augment the Matrix with the Identity Matrix

Step 2: Perform Elementary Row Operations

1. Make the pivot in the first row equal to 1

2. Eliminate the first column in rows 2 and 3

3. Subtract row 2 from row 3 to eliminate the second column in row 3

4. Make the pivot in row 3 equal to 1

5. Eliminate the third column in rows 1 and 2

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Chapter Questions
54 MCQs