Question

Let Aequals=[Start 2 By 2 Matrix 1st Row 1st Column 4 2nd Column 2 2nd Row 1st Column negative 1 2nd Column 2 EndMatrix ]4 2 −1 2 and Bequals=[Start 2 By 2 Matrix 1st Row 1st Column 2 2nd Column 4 2nd Row 1st Column negative 2 2nd Column k EndMatrix ]2 4 −2 k . What​ value(s) of​ k, if​ any, will make ABequals=​BA?

Answer 1

`k=-2`

Explanation:

Given matrix

`A=\left[\begin{array}{ccc}4&2\\-1&2\end{array}\right]`

And

`B=\left[\begin{array}{ccc}2&4\\-2&k\end{array}\right]`

We need to find the value of
`k`, that will make
`AB=BA`.

Let us find
`AB`

`AB=\left[\begin{array}{ccc}4&2\\-1&2\end{array}\right]* \left[\begin{array}{ccc}2&4\\-2&k\end{array}\right]`

`AB=\left[\begin{array}{ccc}4&16+2k\\-6&-4+2k\end{array}\right]`

Now, let us find
`BA`

`BA=\left[\begin{array}{ccc}2&4\\-2&k\end{array}\right]* \left[\begin{array}{ccc}4&2\\-1&2\end{array}\right]`

`BA=\left[\begin{array}{ccc}4&12\\-8-k&-4+2k\end{array}\right]`

Put
`AB=BA`

`\left[\begin{array}{ccc}4&16+2k\\-6&-4+2k\end{array}\right]=\left[\begin{array}{ccc}4&12\\-8-k&-4+2k\end{array}\right]`

In order to make
`AB=BA`, the
`16+2k=12\ and\ -6=-8-k` should be equal. As other two values are equal that are
`4\ and -4+2k`

By solving each equation

`16+2k=12\\2k=-4\\k=-2`

And

`-6=-8-k\\-k=-6+8\\k=-2`

We can see if the value
`k=-2` then
`AB=BA`