Matrices and Linear Algebra

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

Which of the following matrices is invertible?

If \( A \) and \( B \) are \( 2 \times 2 \) matrices such that \( AB = BA \), which of the following must be true?

Find the inverse of \( A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} \).

If \( A = \begin{bmatrix}1 & 0 & 2 \\ 0 & 1 & -1 \\ 3 & 0 & 1\end{bmatrix} \), what is \( \text{rank}(A) \)?

Solve the system of equations using matrix method: \( 2x + 3y = 5 \), \( 4x + 6y = 10 \). What is the nature of the system?

Let \( A \) be a \( 3 \times 3 \) matrix with eigenvalues 2, 3, and 5. What is \( \det(2A) \)?

Given \( A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} \), find \( A^2 - 5A + 6I \).

If \( A \) is a \( 2 \times 2 \) matrix such that \( A^2 = I \) and \( \det(A) = -1 \), what is the trace of \( A \)?

For the matrix \( A = \begin{bmatrix}4 & 1 & 2 \\ 0 & 3 & -1 \\ 0 & 0 & 5\end{bmatrix} \), find \( \det(A) \).