Answer
The matrix is invertible.
Work Step by Step
We are given the matrix $\begin{bmatrix}
5 & 1 & -1 \\
1 & -3 &-2 \\
0 & 5 & 3\\
\end{bmatrix}$.
To find the determinant:
$\begin{vmatrix}
5 & 1 & -1 \\
1 & -3 &-2 \\
0 & 5 & 3\\
\end{vmatrix}$ = 5 $\begin{vmatrix} -3 & -2\\
5 & 3\\
\end{vmatrix}$ -1 $\begin{vmatrix} 1 & -1\\
5 & 3\\
\end{vmatrix}$ +0 $\begin{vmatrix} 1 & -1\\
-3 & -2\\
\end{vmatrix}$
= 5 (-9+10) - (3+5) +0
=5 - 8
= -3
Thus, since the determinant of the matrix does not equal zero, the matrix is invertible.
