Understanding Matrices: Operations and Applications

Given two $n imes n$ matrices, $A$ and $B$, under what condition is it always true that $(A + B)^2 = A^2 + 2AB + B^2$?

Consider matrix $A$, a square matrix with a determinant of zero. Which of the following statements must be true?

If a matrix $A$ is symmetric, what can be definitively stated about its eigenvalues?

If matrix $P$ is used for encoding messages and its inverse $P^{-1}$ for decoding, what condition must be strictly met for perfect decoding?

For a matrix to be considered diagonal, which condition must its elements satisfy?

What is the rank of an $n imes n$ identity matrix?

When a matrix is in reduced row echelon form, what is true about the leading entry in each non-zero row?

What is the primary objective of applying Gaussian elimination to a matrix?

In the context of LU factorization, what type of matrix is 'L'?

What is the main purpose of using Successive Over-Relaxation (SOR) in iterative methods?

Suppose matrix $A$ is a $5 \times 5$ matrix with a determinant of 3. If matrix $B$ is formed by multiplying the second row of $A$ by 2 and the fourth row by 5, what is the determinant of matrix $B$?

Given matrix $M = \begin{bmatrix} 1 & 2 \ 2 & 4 \end{bmatrix}$, which of the following statements is correct?

Suppose $A$ and $B$ are $n \times n$ matrices and $AB = I$ (the identity matrix). Which of the following MUST be true?

Which of the following is a property of orthogonal matrices?

Consider a linear transformation in 3D space represented by a $3 \times 3$ matrix. If the transformation scales space by a factor of 2 in all directions, what is the determinant of this matrix?

If a square matrix $A$ has an eigenvalue of 0, what does this imply about the matrix?

For which type of matrix is LU decomposition not directly applicable without pivoting?

If a matrix $A$ is known to be symmetric positive definite, which of the following statements is guaranteed to be true?

How does the Successive Over-Relaxation (SOR) method modify the Gauss-Seidel method to potentially improve convergence?

Let $A$ be a $3 \times 3$ matrix with eigenvalues $\lambda_1 = 1$, $\lambda_2 = -1$, and $\lambda_3 = 2$. What is the determinant of $A^2$?

Given a square matrix $A$, if $A^2 = A$, what are the possible eigenvalues of $A$?

Suppose $A$ is a real $n \times n$ matrix. Which of the following guarantees that $A$ is diagonalizable?

Consider the matrix $A = \begin{bmatrix} 2 & -1 \ -1 & 2 \end{bmatrix}$. What are its eigenvalues?

If $A$ is an $n \times n$ matrix with determinant 5, what is the determinant of $2A$?

Let $A$ and $B$ be two $n imes n$ matrices. Which of the following statements is NOT always true?

Suppose the matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$ has determinant 1 and trace 3. What is the trace of $A^{-1}$?

Given that $A$ is an $n imes n$ matrix, which statement is true regarding the number of linearly independent eigenvectors associated with a repeated eigenvalue $\lambda$?

Which of the following is the correct formula for calculating the determinant of a $3 \times 3$ matrix $A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$?

Let $A$ be a real $n \times n$ matrix such that $A^T = -A$. Which of the following statements is always true?

Consider a matrix $A$ representing a linear transformation. If the column space of $A$ is equal to the null space of $A$, which of the following must be true?