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

Answer:

$k=-2$

Step-by-step 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 $AB$

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

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

Now, let us find $BA$

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

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

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 $AB=BA$