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\\eq0`, as


`\det\mathbf A=\begin{vmatrix}1&7\\0&1\end{vmatrix}=1*1-0*7=1`

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


`\mathbf A^(-1)=\frac1{\det\mathbf A}\begin{bmatrix}1&0\\-7&1\end{bmatrix}^\top=\begin{bmatrix}1&-7\\0&1\end{bmatrix}`