Matrices And Determinants

To determine whether the determinants of the given matrices are even numbers, we need to calculate the determinant of each. The determinant of a 2x2 matrix is computed using the formula:
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Let's compute the determinant for each matrix:

For matrix :
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The determinant is 4, which is even.

For matrix :
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The determinant is 194, which is even.

For matrix :
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The determinant is 251, which is odd.

For matrix :
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The determinant is 222, which is even.

Thus, the matrices for which the determinant is an even number are: options (A), (B), and (D). Therefore, the correct answer is: (A), (B) and (D) only

Step 1: Understanding the Concept:

We need to identify the incorrect statement among the given properties of invertible matrices. An invertible matrix is a square matrix that has a non-zero determinant.

Step 3: Detailed Explanation:

Let's analyze each statement.

1. :

The formula for the inverse of a matrix is given by: Multiplying both sides by (which is non-zero since is invertible), we get: This statement is correct.

2. :

This property states that the inverse of a sum is the sum of the inverses. This is generally not true for matrices. We can show this with a counterexample.

Let and (where is the identity matrix). Both are invertible.

LHS = .

RHS = .

Since LHS RHS, the statement is NOT correct.

3. :

We know that a matrix and its inverse satisfy the relation .

Taking the determinant of both sides: Using the property , we get: Since is invertible, . We can divide by : This statement is correct.

4. :

This is the well-known reversal law for the inverse of a product of matrices. It is a standard property.

This statement is correct.

Step 4: Final Answer:

The statement that is not correct is .

Step 1: Understanding the Concept:

This problem involves simplifying a matrix polynomial expression using the properties of matrix algebra and a given condition . Since a matrix and the identity matrix commute (i.e., ), we can use standard binomial expansion formulas.

Step 2: Key Formula or Approach:

We use the binomial expansion formulas:

Step 3: Detailed Explanation:

Let's expand the terms and :

Since and , this becomes:

Now, let's expand the second term:

Now add the two expansions:

We are given that . Let's find an expression for :

Substitute back into the expression:

Finally, subtract the last term from the original question:

Step 4: Final Answer:

The value of the expression is 5A.

Step 1: Understanding the Concept:

This question involves properties of matrix multiplication and determinants. We need to determine the size of the product matrix and then apply the property for the determinant of a scalar multiple of a matrix.

Step 2: Key Formula or Approach:

1. Determine the order (dimensions) of the product matrix .
2. Use the property of determinants: For a square matrix of order and a scalar , .

Step 3: Detailed Explanation:

First, let's find the order of the product matrix .

The order of matrix is .
The order of matrix is .
For the product to be defined, the number of columns in must equal the number of rows in . Here, it is 3, so the product is defined.
The order of the resulting matrix is (number of rows of ) (number of columns of ), which is .
So, is a square matrix of order .

Now we need to find the determinant of . We use the property .

Here, , , and the order .

Note that options (1) and (2) are incorrect because the determinant is only defined for square matrices. Since (2x3) and (3x2) are not square, and are not defined.

Step 4: Final Answer:

The value of is .

Step 1: Understanding the Concept:

This problem involves simplifying a matrix polynomial expression using the properties of matrix algebra and a given condition . Since a matrix and the identity matrix commute (i.e., ), we can use standard binomial expansion formulas.

Step 2: Key Formula or Approach:

We use the binomial expansion formulas:

Step 3: Detailed Explanation:

Let's expand the terms and :

Since and , this becomes:

Now, let's expand the second term:

Now add the two expansions:

We are given that . Let's find an expression for :

Substitute back into the expression:

Finally, subtract the last term from the original question:

Step 4: Final Answer:

The value of the expression is 5A.

Step 1: Understanding the Concept:

We need to calculate specific minors and cofactors of the given 3x3 matrix and check which of the given statements are true.

The minor of an element is the determinant of the submatrix formed by deleting the -th row and -th column.

The cofactor is related to the minor by the formula .

Step 2: Key Formula or Approach:

We will use the above definitions and calculate the minors and cofactors for each statement.

Step 3: Detailed Explanation:

The given matrix is: Let's evaluate each statement.

(A) :
is the minor of the element . We delete the 2nd row and 2nd column. Statement (A) is true.

(B) :
is the cofactor of the element . First, find the minor . Now, calculate the cofactor. Statement (B) is true.

(C) :
is the cofactor of the element . First, find the minor . Now, calculate the cofactor. Statement (C) is false.

(D) :
From our calculation for statement (B), we found that . Statement (D) is false.

(E) :
The OCR here is likely a typo. Based on the options, let's assume it should have been , or something related. From our calculation for statement (C), we found . Statement (E) is false.

Step 4: Final Answer:

The only true statements are (A) and (B). Therefore, the correct option is (1).

Step 1: Understanding the Concept:

We are given a matrix equation involving a matrix , its transpose , and the identity matrix . We need to solve this equation to find the general value of the angle .

Step 2: Key Formula or Approach:

1. Find the transpose of matrix , .
2. Calculate the sum .
3. Set this sum equal to the identity matrix .
4. Solve the resulting trigonometric equation for .

Step 3: Detailed Explanation:

The given matrix is .

First, find the transpose of , which is obtained by interchanging rows and columns:

Now, add and :

We are given that :

By equating the corresponding elements of the matrices, we get the equation:

The principal value for is .

The general solution for is , where is an integer.

So, the general solution for is , where .

Looking at the options provided, the option is one part of the general solution. It is the most appropriate choice among the given options.

Step 4: Final Answer:

The value of is given by , where .

Step 1: Understanding the Concept:

A matrix is symmetric if .
A matrix is skew-symmetric if .
We are given that and are skew-symmetric, so and . We need to check the properties of combinations of these matrices. A useful property is that for a skew-symmetric matrix , is symmetric if is even, and skew-symmetric if is odd.

Step 2: Key Formula or Approach:

We will use the properties of matrix transposition and the fact that for skew-symmetric matrices, the power determines the symmetry of the matrix.

Step 3: Detailed Explanation:

1. is skew-symmetric:

Let . Let's find its transpose:

Since and : Since , the matrix is skew-symmetric. This statement is true.

2. is skew-symmetric:

Let . The power 19 is odd:

The matrix is skew-symmetric. This statement is true.

3. is symmetric:

Let . The power 14 is even:

The matrix is symmetric. This statement is true.

4. is symmetric:

Let . Let's find its transpose:

For to be symmetric, must be equal to : This implies , or , which means must be the zero matrix. This is not true for all skew-symmetric matrices . Therefore, is not generally symmetric. This statement is NOT true.

Step 4: Final Answer:

The statement that is not true is is symmetric.

Step 1: Understanding the Concept

The task is to find the inverse of each matrix in List-I and match it with the correct inverse from List-II.

Step 2: Key Formula or Approach

For a matrix , the inverse is:

Step 3: Detailed Explanation

(A) For :
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This matches (III).

(B) For :
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This matches (I).

(C) For :
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This matches (IV).

(D) For :
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This matches (II).

Step 4: Final Answer

→ Option (C).

Step 1: Understanding the Concept:

This question involves properties of matrix multiplication and determinants. We need to determine the size of the product matrix and then apply the property for the determinant of a scalar multiple of a matrix.

Step 2: Key Formula or Approach:

1. Determine the order (dimensions) of the product matrix .
2. Use the property of determinants: For a square matrix of order and a scalar , .

Step 3: Detailed Explanation:

First, let's find the order of the product matrix .

The order of matrix is .
The order of matrix is .
For the product to be defined, the number of columns in must equal the number of rows in . Here, it is 3, so the product is defined.
The order of the resulting matrix is (number of rows of ) (number of columns of ), which is .
So, is a square matrix of order .

Now we need to find the determinant of . We use the property .

Here, , , and the order .

Note that options (1) and (2) are incorrect because the determinant is only defined for square matrices. Since (2x3) and (3x2) are not square, and are not defined.

Step 4: Final Answer:

The value of is .

Step 1: Understanding the Concept:

This question tests the knowledge of basic definitions related to matrices, specifically symmetric, skew-symmetric, singular, and non-singular matrices.

Step 3: Detailed Explanation:

Let's analyze each item in List-I and match it with the correct definition in List-II.

(A) : This is the definition of a symmetric matrix. A matrix is symmetric if it is equal to its transpose. This matches with (IV).

(B) : This is the definition of a skew-symmetric matrix. A matrix is skew-symmetric if its transpose is equal to its negative. This matches with (III).

(C) : The determinant of a matrix being zero is the condition for the matrix to be a singular matrix. This matches with (I).

(D) : The determinant of a matrix being non-zero is the condition for the matrix to be a non-singular matrix. Such matrices have an inverse. This matches with (II).

Step 4: Final Answer:

Combining the matches, we get:

• (A) → (IV)
• (B) → (III)
• (C) → (I)
• (D) → (II)

This combination corresponds to option (2).

Step 1: Understanding the Concept:

We need to identify the incorrect statement among the given properties of invertible matrices. An invertible matrix is a square matrix that has a non-zero determinant.

Step 3: Detailed Explanation:

Let's analyze each statement.

1. :

The formula for the inverse of a matrix is given by: Multiplying both sides by (which is non-zero since is invertible), we get: This statement is correct.

2. :

This property states that the inverse of a sum is the sum of the inverses. This is generally not true for matrices. We can show this with a counterexample.

Let and (where is the identity matrix). Both are invertible.

LHS = .

RHS = .

Since LHS RHS, the statement is NOT correct.

3. :

We know that a matrix and its inverse satisfy the relation .

Taking the determinant of both sides: Using the property , we get: Since is invertible, . We can divide by : This statement is correct.

4. :

This is the well-known reversal law for the inverse of a product of matrices. It is a standard property.

This statement is correct.

Step 4: Final Answer:

The statement that is not correct is .

Step 1: Understanding the Concept:

A matrix is symmetric if .
A matrix is skew-symmetric if .
We are given that and are skew-symmetric, so and . We need to check the properties of combinations of these matrices. A useful property is that for a skew-symmetric matrix , is symmetric if is even, and skew-symmetric if is odd.

Step 2: Key Formula or Approach:

We will use the properties of matrix transposition and the fact that for skew-symmetric matrices, the power determines the symmetry of the matrix.

Step 3: Detailed Explanation:

1. is skew-symmetric:

Let . Let's find its transpose:

Since and : Since , the matrix is skew-symmetric. This statement is true.

2. is skew-symmetric:

Let . The power 19 is odd:

The matrix is skew-symmetric. This statement is true.

3. is symmetric:

Let . The power 14 is even:

The matrix is symmetric. This statement is true.

4. is symmetric:

Let . Let's find its transpose:

For to be symmetric, must be equal to : This implies , or , which means must be the zero matrix. This is not true for all skew-symmetric matrices . Therefore, is not generally symmetric. This statement is NOT true.

Step 4: Final Answer:

The statement that is not true is is symmetric.