# Math Help - [SOLVED] 2 Matrix questions, Invertible and A^2=A

1. ## [SOLVED] 2 Matrix questions, Invertible and A^2=A

I have no idea what to do for either of these questions

1. If A = |x -1|
|y -1|

determine the value of x and y such that A^2=A

2. The matrix |2 -9|
|9 k|

is invertible if and only if k !=??

2. 1. $

A^2=A \rightarrow A^2-A=(A-I_2)A=0
\rightarrow im(A)\in \ker(A-I_2)
$

$
\rightarrow im(A)=span(\begin{bmatrix}x\\y\end{bmatrix},
\begin{bmatrix}-1 \\-1 \end{bmatrix})
\rightarrow
\left.\begin{array}{cc}
(A-I_2)\begin{bmatrix}x\\y\end{bmatrix}=0 \\
(A-I_2)\begin{bmatrix}-1 \\-1 \end{bmatrix}=0 \\
\end{array}\right\} \\
\rightarrow x=y=2

$

2. $\det\begin{bmatrix}2&-9\\9&k\end{bmatrix}=
2k+81\neq 0 \rightarrow k\neq -\frac{81}{2}
$

3. thanks