## Find Inverse Matrices Using Adjoint Matrices

## Problem 546

Let $A$ be an $n\times n$ matrix.

The $(i, j)$ **cofactor** $C_{ij}$ of $A$ is defined to be

\[C_{ij}=(-1)^{ij}\det(M_{ij}),\]
where $M_{ij}$ is the $(i,j)$ **minor matrix** obtained from $A$ removing the $i$-th row and $j$-th column.

Then consider the $n\times n$ matrix $C=(C_{ij})$, and define the $n\times n$ matrix $\Adj(A)=C^{\trans}$.

The matrix $\Adj(A)$ is called the **adjoint** matrix of $A$.

When $A$ is invertible, then its inverse can be obtained by the formula

For each of the following matrices, determine whether it is invertible, and if so, then find the invertible matrix using the above formula.

**(a)** $A=\begin{bmatrix}

1 & 5 & 2 \\

0 &-1 &2 \\

0 & 0 & 1

\end{bmatrix}$.

**(b)** $B=\begin{bmatrix}

1 & 0 & 2 \\

0 &1 &4 \\

3 & 0 & 1

\end{bmatrix}$.