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
ANSWERED
asked 2021-05-27
Find k such that the following matrix M is singular.
\(M=\begin{bmatrix}-1 & -1 & -2\\ 0 & -1 & -4 \\ -12+k & -2 & -2 \end{bmatrix}\)
\(k=?\)

Expert Answers (1)

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 \(k=9\)
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