Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation
Problem 289
(a) Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 0 & 1 \\
1 &0 &0 \\
2 & 1 & 1
\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\]
where
\[B=\begin{bmatrix}
4 & 1\\
2& 6
\end{bmatrix}.\]
Verify that the matrix $A$ you obtained is actually a nonsingular matrix.
(The Ohio State University, Linear Algebra Midterm Exam Problem)
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