Answer to Question #124666 in Linear Algebra for desmond

Question

(b) If
A =


4 4 −2
−1 0 1
3 6 −1

 ,
show that A2 = A + kI for some constant k, where I is the unit matrix of order 3. Hence
find the inverse matrix A−1
. [

Answer 1

Given, $A =\begin{bmatrix} 4 & 4 & -2 \\ -1 & 0 & 1\\ 3 & 6 & -1 \end{bmatrix}$.

$A^{2}=\begin{bmatrix} 4 & 4 & -2 \\ -1 & 0 & 1\\ 3 & 6 & -1 \end{bmatrix} \cdot \begin{bmatrix} 4 & 4 & -2 \\ -1 & 0 & 1\\ 3 & 6 & -1 \end{bmatrix}\\~\\~~~~~~ = \begin{bmatrix} 6 & 4 & -2 \\ -1 & 2 & 1\\ 3 & 6 & 1 \end{bmatrix}\\~\\ ~~~~~~= \begin{bmatrix} 4 & 4 & -2 \\ -1 & 0 & 1\\ 3 & 6 & -1 \end{bmatrix} + \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0\\ 0 & 0 & 2 \end{bmatrix}\\~\\~~~~~~ = A + 2I.$

Therefore, $A^{2}=A+kI$ for $k=2$.

Multiplying, $A^{2}=A+2I$ by $A^{-1}$, we get

$A = I + 2A^{-1}\\~\\ A^{-1} = \frac{1}{2}(A-I)\\~\\ ~~~~~~~~=\dfrac{1}{2}\begin{bmatrix} 3 & 4 & 2\\ -1 & -1 & 1 \\3 & 6 & -2\end{bmatrix}$