其中 $p$ 和 $q$ 是非零实常数。

(a) 用 $p$ 和 $q$ 表示求 $A^{-1}$。

已知 $XA = B$，其中

(b) 求矩阵 $X$，将答案写成最简形式。

先计算 $A$ 的行列式。由于 $p,q$ 均非零，行列式 $pq$ 非零，因此逆矩阵存在；再交换主对角线元素并改变副对角线元素的符号。

The determinant of $A$ is

Since $p$ and $q$ are non-zero, $\det(A)\neq0$. Therefore,

由 $XA=B$ 在右边乘 $A^{-1}$，得到 $X=BA^{-1}$。矩阵乘法时逐行乘逐列，$p,q$ 会全部约去。

Since $XA=B$,

Thus,

把 $X$ 的六个元素设为未知数，直接展开 $XA=B$。每一行产生一组二元一次方程，分别求解三组即可。

Let

Since $p$ and $q$ are non-zero, the factors $p$ in the first column and $q$ in the second column can be cancelled. Thus $XA=B$ gives three pairs of simultaneous equations.

For the first row,

Multiplying the first equation by $3$ and the second by $2$ gives

Subtracting gives $b=-1$. Substituting into the first equation,

For the second row,

Multiplying these equations by $3$ and $2$, respectively, gives

Subtracting gives $d=4$. Therefore,

For the third row,

Multiplying these equations by $3$ and $2$, respectively, gives

Subtracting gives $f=1$. Therefore,

Hence,