Question

Find a 2 × 2 matrix b ≠ o and b ≠ i2 such that ab = ba, where a = 1 7 0 1 . b =

Answer 1

If $\mathbf A$ is invertible, then we can simply choose $\mathbf B=\mathbf A^{-1}$ because for any invertible matrix $\mathbf X$ by definition we must have $\mathbf{XX}^{-1}=\mathbf X^{-1}\mathbf X=\mathbf I$.

We have $\det\mathbf A\neq0$, as

$\det\mathbf A=\begin{vmatrix}1&7\\0&1\end{vmatrix}=1\times1-0\times7=1$

which means $\mathbf A$ is non-singular and has an inverse. The inverse itself would be

$\mathbf A^{-1}=\dfrac1{\det\mathbf A}\begin{bmatrix}1&0\\-7&1\end{bmatrix}^\top=\begin{bmatrix}1&-7\\0&1\end{bmatrix}$