Matrix algebra integration: Practice Questions

Given the matrix \( M = \begin{pmatrix} k & 2 \\ 3 & 4 \end{pmatrix} \), find the value of \( k \) such that the determinant of \( M \) is zero.

If \( AB = C \), where \( A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \) and \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \), find the matrix \( B \).

The matrix \( N = \begin{pmatrix} a-2 & 4 \\ 3 & a+2 \end{pmatrix} \) is singular. Determine the two possible values of the real constant \( a \).

Find the determinant of the matrix \( A = \begin{pmatrix} 4 & -2 \\ 5 & 3 \end{pmatrix} \).

Given the matrices \( A = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix} \) and \( B = \begin{pmatrix} 0 & 2 \\ 4 & -1 \end{pmatrix} \), calculate the matrix product \( AB \).

Calculate the determinant of the matrix \( \mathbf{A} = \begin{pmatrix} 4 & -2 \\ 3 & 5 \end{pmatrix} \).

Find the value of the constant \( k \) for which the matrix \( \mathbf{M} = \begin{pmatrix} k & 4 \\ 3 & k-1 \end{pmatrix} \) is singular.

The matrix \( {M} \) is defined by \( {M} = \begin{pmatrix} k & 2 \\ 3 & k-1 \end{pmatrix} \), where \(k\) is a real constant.
(a) Find the values of \(k\) for which \( {M} \) is a singular matrix.
(b) Given that \(k = 4\), find \( {M}^{-1} \).