Question

# Find k such that the following matrix M is singular. M=\begin{bmatrix}-1 & -1 & -2\\ 0 & -1 & -4 \\ -12+k & -2 & -2 \end{bmatrix} k=?

Matrix transformations
Find k such that the following matrix M is singular.
$$M=\begin{bmatrix}-1 & -1 & -2\\ 0 & -1 & -4 \\ -12+k & -2 & -2 \end{bmatrix}$$
$$k=?$$

2021-05-28
Step 1
Consider the matrix: $$M=\begin{bmatrix}-1 & -1 & -2\\ 0 & -1 & -4 \\ -12+k & -2 & -2 \end{bmatrix}$$
A square matrix is singulare if and only if it's determinant is 0
Now, the determinant is $$\begin{bmatrix}-1 & -1 & -2\\ 0 & -1 & -4 \\ -12+k & -2 & -2 \end{bmatrix}=0$$
$$(-1)\begin{bmatrix}-1 & -4 \\ -2 & -2 \end{bmatrix}-(-1)\begin{bmatrix}0 & -4 \\ -12+k & -2 \end{bmatrix}+(-2)\begin{bmatrix}0 & -1 \\ -12+k & -2 \end{bmatrix}=0$$
$$(-1)(2-8)+1(0+4(-12+k))-2(0+1(-12+k))=0$$
$$6+4(-12+k)-2(-12+k)=0$$
$$6-48+4k+24-2k=0$$
$$-18+2k=0$$
$$2k=18$$
Hence, the required value of k for the singular matrix M is $$k=9$$