Find all Values of x such that the Given Matrix is Invertible

Stanford University Linear Algebra Exam Problems and Solutions

Problem 32

Let
\[ A=\begin{bmatrix}
2 & 0 & 10 \\
0 &7+x &-3 \\
0 & 4 & x
\end{bmatrix}.\] Find all values of $x$ such that $A$ is invertible.

(Stanford University Linear Algebra Exam)

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Hint.

Calculate the determinant of the matrix $A$.

Solution.

A matrix is invertible if and only if its determinant is non-zero.
So we first calculate the determinant of the matrix $A$.

By the first column cofactor expansion, we have

\begin{align*}
\det(A) &=2 \begin{vmatrix}
7+x & -3\\
4& x
\end{vmatrix} \\
&=2\left( (7+x)x-(-3)4 \right)=2(x^2+7x+12)\\
&=2(x+3)(x+4).
\end{align*}

Thus the determinant of $A$ is zero if and only if $x=-3$ or $x=-4$.
Therefore the matrix $A$ is invertible for all $x$ except $x=-3$ and $x=-4$.

Stanford University Linear Algebra Exam Problems and Solutions

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  1. 07/19/2017

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