Matrices

If \( A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \), what is the determinant of matrix \( A \)?

If \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \), what is \( AB \)?

Which of the following matrices is symmetric?

If \( A = \begin{bmatrix} 3 & 0 \\ 4 & 5 \end{bmatrix} \), what is \( A^{-1} \)?

Let \( A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix} \). What is the trace of \( A \)?

If \( A \) is a \( 3 \times 3 \) matrix such that \( A^2 = I \) (identity matrix), which of the following must be true?

Find the rank of matrix \( M = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} \).

If \( A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \), find \( AB - BA \).

If \( A \) is a \( 2 \times 2 \) matrix with \( \det(A) = 3 \) and \( \det(B) = 5 \), what is \( \det(3AB) \)?

Given \( A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 3 \\ 4 & 0 & 1 \end{bmatrix} \), find \( \det(A) \).