Find k such that the following matrix M is singular. \[M=\begin{bmatrix}-4&0&-3\\-4&-4&1\\-14+k&-8&-1\end{bmatrix}\]

Jason Watson 2021-11-19 Answered
Find k such that the following matrix M is singular.
\[M=\begin{bmatrix}-4&0&-3\\-4&-4&1\\-14+k&-8&-1\end{bmatrix}\]

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Expert Answer

Charles Clute
Answered 2021-11-20 Author has 505 answers
A square matrix A is said to be singular if its determinant value is 0.
i.e \(\displaystyle{\left|{A}\right|}={0}\)
Here, the given matrix is \[M=\begin{bmatrix}-4&0&-3\\-4&-4&1\\-14+k&-8&-1\end{bmatrix}\]
So we must have
\(\displaystyle{\left|{M}\right|}={0}\)
\[\begin{bmatrix}-4&0&-3\\-4&-4&1\\-14+k&-8&-1\end{bmatrix}=0\]
Now, expanding the determinant by its 1st row
\(\displaystyle-{4}{\left({4}+{8}\right)}-{0}-{3}{\left({32}+{4}{\left(-{14}+{k}\right)}\right)}={0}\)
\(\displaystyle-{48}-{3}{\left({32}-{96}+{4}{k}\right)}={0}\)
\(\displaystyle-{48}-{3}{\left(-{64}+{4}{k}\right)}={0}\)
\(\displaystyle-{48}+{192}-{12}{k}={0}\)
\(\displaystyle{144}-{12}{k}={0}\)
\(\displaystyle{12}{k}={144}\)
\(\displaystyle{k}={12}\)
So ,the required value of k=12 for which the matric is singular.
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