Linear Algebra

Step 1: Understanding the Concept:
According to the Cayley-Hamilton Theorem, every square matrix satisfies its own characteristic equation. We can find the characteristic equation of the matrix A and compare it with the given polynomial equation to find the values of and .

Step 2: Key Formula or Approach:
The characteristic equation of a matrix A is given by . Since A is an upper triangular matrix, its eigenvalues are its diagonal entries. The eigenvalues are . The characteristic polynomial is .
Step 3: Detailed Explanation:
By the Cayley-Hamilton Theorem, the matrix A must satisfy this equation: We are given the equation: Comparing the coefficients of the two equations, we get:
Step 4: Final Answer:
The respective values are and .

Step 1: Understanding the Concept:
A matrix M is symmetric if it is equal to its transpose, i.e., . We are given that A and B are symmetric, so and . We need to check the validity of the given statements based on this property.

Step 2: Key Formula or Approach:
We will use the properties of transpose: and .

Step 3: Detailed Explanation:

A. AB is symmetric: For AB to be symmetric, we must have . Using the transpose property, . Since B and A are symmetric, and . So, . Therefore, AB is symmetric if and only if , i.e., if A and B commute. This is not true for all symmetric matrices. So, statement A is incorrect.
B. A+B is symmetric: For A+B to be symmetric, we must have . Using the transpose property, . Since A and B are symmetric, and . So, . This is always true. So, statement B is correct.
C. : We are given that A and B are symmetric, so and . Substituting these into the statement gives . This is a tautology (always true). So, statement C is correct.
D. : Let's evaluate the right side. Using the transpose property, . Since A and B are symmetric, this becomes . So the statement is , which is a tautology (always true). So, statement D is correct.
Step 4: Final Answer:
The correct statements are B, C, and D. Statement A is not always true.

Step 1: Understanding the Concept:
This question is about basic feasible solutions (BFS) in the context of linear algebra and linear programming. A basic solution to a system of linear equations is found by setting a certain number of variables to zero and solving for the rest.
Note on a typo in the question: For the matrix product AX to be defined and equal to b, if A is and X is , then b must be an matrix. The question states b is , which is a typo. We proceed assuming b is .

Step 2: Key Definitions:
- The system has linear equations and variables. - We are given that the rank of the matrix A is , which implies and that the equations are linearly independent. - A basic solution is obtained by: 1. Choosing variables out of the variables to be "basic variables". 2. Setting the remaining variables ("non-basic variables") to zero. 3. Solving the resulting system for the basic variables. - The selection of basic variables is valid if the corresponding columns of the matrix A are linearly independent, forming an invertible submatrix. - A basic feasible solution (BFS) is a basic solution where all the variables (specifically, the basic ones) are non-negative.

Step 3: Detailed Explanation:
The question asks for the maximum number of basic feasible solutions. The number of BFS depends on the specific values in A and b. However, the theoretical maximum is limited by the total number of possible basic solutions. A basic solution is determined by the choice of which variables are basic. Since the rank of A is , A has at least one set of linearly independent columns. The maximum number of ways to form a basic solution is the number of ways to choose columns from the available columns. This number is given by the binomial coefficient "n choose m": Since every BFS is, by definition, a basic solution, the number of BFS cannot exceed the total number of basic solutions. Therefore, the maximum possible number of basic feasible solutions is .

Step 4: Final Answer:
The maximum number of basic feasible solutions for the given system is .

Step 1: Understanding the Concept:
A set of vectors in an -dimensional space (here, 3 vectors in ) is linearly dependent if and only if the determinant of the matrix formed by these vectors as columns (or rows) is zero.

Step 2: Key Formula or Approach:
We will form a 3x3 matrix using the given vectors as columns and set its determinant to zero to solve for . We need to solve .

Step 3: Detailed Explanation:
We calculate the determinant of , for instance, by cofactor expansion along the third column, as it contains a zero which simplifies the calculation. For the vectors to be linearly dependent, the determinant must be zero:
Step 4: Final Answer:
The value of for which the vectors are linearly dependent is 1.

Step 1: Understanding the Concept:
A system of linear equations has a unique solution if and only if the determinant of the coefficient matrix A is non-zero. The values on the right-hand side of the equations (in this case, ) do not affect the uniqueness of the solution, only its existence.

Step 2: Key Formula or Approach:
We write the system in matrix form and find the determinant of the coefficient matrix A. For a unique solution, we must have .

Step 3: Detailed Explanation:
Let's calculate the determinant of A. For the system to have a unique solution, we need . If , the system has a unique solution regardless of the value of . The value of can be any real number.

Step 4: Final Answer:
The system has a unique solution if and .

Step 1: Understanding the Concept:
This question relates the trace and determinant of a 2x2 matrix to the trace of its square. We can use the Cayley-Hamilton theorem, which states that a matrix satisfies its own characteristic equation.

Step 2: Key Formula or Approach:
For a matrix A, the characteristic equation is given by: By the Cayley-Hamilton theorem, the matrix A satisfies this equation: We can rearrange this equation to find an expression for and then take the trace.

Step 3: Detailed Explanation:
Substitute the given values into the Cayley-Hamilton equation: Rearrange to solve for : Now, take the trace of both sides of the equation. We use the properties that and . We are given . The trace of the identity matrix is . Alternative Method using Eigenvalues: Let the eigenvalues of A be and . Then: - - The eigenvalues of are and . - We can find from the known sums and products:
Step 4: Final Answer:
The value of is 17.

Step 1: Understanding the Concept:
A set of vectors forms a basis for a vector space if two conditions are met: 1. The vectors are linearly independent. 2. The vectors span the vector space. For the vector space , which has a dimension of 3, any set of 3 linearly independent vectors will form a basis. Any set with fewer than 3 vectors cannot span , and any set with more than 3 vectors must be linearly dependent.

Step 2: Key Formula or Approach:
We can eliminate options based on the number of vectors. For the options with exactly 3 vectors, we can check for linear independence by forming a matrix with the vectors as rows or columns and calculating its determinant. If the determinant is non-zero, the vectors are linearly independent and thus form a basis for .

Step 3: Detailed Explanation:

Option A: . This set has only 2 vectors. It cannot span the 3-dimensional space . Not a basis.
Option C: . This set has 4 vectors in a 3-dimensional space. This set must be linearly dependent. Not a basis.
Option B: . This set has 3 vectors. We check for linear independence by calculating the determinant of the matrix formed by these vectors. Since the determinant is , the vectors are linearly independent and form a basis for .
Option D: . This set has 3 vectors. Let's check the determinant. Since the determinant is 0, the vectors are linearly dependent. Not a basis.
Step 4: Final Answer:
Only the set of vectors in option (B) is linearly independent and has the correct number of vectors to form a basis for .

Step 1: Understanding the Concept:
This problem involves the dimensions of subspaces, their sum, and their intersection. We will use the formula for the dimension of the sum of two subspaces.

Step 2: Key Formula or Approach:
The formula relating the dimensions of the sum and intersection of two subspaces U and W is: We are given: - - - - U and W are distinct subspaces.

Step 3: Detailed Explanation:
First, let's rearrange the formula to solve for the dimension of the intersection: Substitute the known values: Now, we need to find the possible values for . - The sum is a subspace of V. Therefore, its dimension cannot exceed the dimension of V: . - Since U and W are subspaces of , the dimension of must be at least as large as the dimension of U and W. So, and . - The problem states that U and W are distinct subspaces. This means . If or , they would not be distinct 4-dimensional subspaces. Since they have the same dimension, one cannot be a proper subset of the other. This implies that must be strictly greater than (which is 4). So, the possible integer values for are 5 and 6. Let's find the corresponding values for : - If , then . - If , then . Therefore, the possible dimensions of are 2 or 3.

Step 4: Final Answer:
The possible dimensions of the intersection are 2 or 3.

Step 1: Understanding the Concept:
A transformation between two vector spaces V and W is linear if it satisfies two conditions for all vectors and any scalar : 1. Additivity: 2. Homogeneity: A quick check is to see if . If not, the transformation cannot be linear.

Step 2: Detailed Explanation:
Let's check each transformation:
A. : Let's check homogeneity. Let and be a scalar. . . Since in general, this is not a linear transformation.
B. : Let's check the zero vector condition. . Since , this is not a linear transformation.
C. : Let's check homogeneity. Let and . . . Since for , this is not a linear transformation.
D. : 1. Additivity: Let and . . . They are equal. Additivity holds. 2. Homogeneity: Let and be a scalar. . . They are equal. Homogeneity holds. Since both conditions are met, this is a linear transformation.
Step 3: Final Answer:
The transformation is the only linear transformation among the options.

Concept 1: Determinant and eigenvalues
The determinant of a matrix is equal to the product of its eigenvalues:
Step 1: Compute determinant of
Given eigenvalues: Concept 2: Determinant of powers
For any square matrix:
Step 2: Apply the formula for

Step 3: Alternative understanding
Eigenvalues of are: Conclusion:

Concept: Order of elements in a cyclic group
In a cyclic group of order , the number of elements of order (where ) is given by Euler’s totient function:
Step 1: Identify the group order

Step 2: Find valid divisors
Possible orders divide 25: We need elements of order .
Step 3: Apply Euler's totient function

Step 4: Interpretation
There are exactly 4 generators of the subgroup of order 5. Conclusion:

Concept: A real symmetric matrix satisfies: This means the entries are symmetric about the main diagonal.
Step 1: Count diagonal elements
There are diagonal elements:
Step 2: Count off-diagonal elements
For , we have: So each pair contributes only one independent element. Number of such pairs:
Step 3: Total independent elements
Hence, the dimension is:

Concept: Symmetric Group
The symmetric group consists of all permutations of elements.
Step 1: Formula for order

Step 2: Apply for

Step 3: Interpretation
There are 6 possible ways to arrange 3 distinct elements. Conclusion:

Concept: Rank of a matrix
The rank of a matrix is the maximum number of linearly independent rows or columns.
Step 1: General rule
For an matrix:
Step 2: Apply to given matrix
Matrix size:
Step 3: Possible values
Rank can be any integer from up to :
Step 4: Interpretation

• Rank : zero matrix
• Rank : full column rank

Conclusion:

Concept: Commutator Subgroup
The commutator of two elements is: The commutator subgroup (or ) is generated by all such commutators.
Step 1: Abelian group definition
A group is Abelian if:
Step 2: Effect on commutators
If , then: (identity element)
Step 3: Conclusion
All commutators are identity commutator subgroup is trivial:

Concept: Rank–Nullity Theorem
This is a fundamental theorem in linear algebra which states:
Step 1: Understand terms

• Rank: Dimension of the image (range) of
• Nullity: Dimension of the kernel (null space) of
• dim( ): Dimension of domain space

Step 2: Interpretation
The theorem splits the domain space into:

• Part mapped to zero (kernel)
• Part mapped to image

Step 3: Conclusion