First slide

Operations of matrix

Question

If $A = [\begin{array}{ccc} 1 & 2 & 1 \\ 0 & 1 & â 1 \\ 3 & â 1 & 1 \end{array}]$ then $A^{3} â 3 A^{2} â A + 9 I$ is equal to

Easy

A

$O$
B

$I$
C

$A$
D

$A^{2}$

Solution

$\begin{aligned} A^{2} = [\begin{array}{ccc} 1 & 2 & 1 \\ 0 & 1 & â 1 \\ 3 & â 1 & 1 \end{array}] [\begin{array}{ccc} 1 & 2 & 1 \\ 0 & 1 & â 1 \\ 3 & â 1 & 1 \end{array}] = [\begin{array}{ccc} 4 & 3 & 0 \\ â 3 & 2 & â 2 \\ 6 & 4 & 5 \end{array}] \\ A^{3} = [\begin{array}{ccc} 4 & 3 & 0 \\ â 3 & 2 & â 2 \\ 6 & 4 & 5 \end{array}] [\begin{array}{ccc} 1 & 2 & 1 \\ 0 & 1 & â 1 \\ 3 & â 1 & 1 \end{array}] = [\begin{array}{ccc} 4 & 11 & 1 \\ â 9 & â 2 & â 7 \\ 21 & 11 & 7 \end{array}] \end{aligned}$
Now, $A^{3} â 3 A^{2} â A + 9 I$
$\begin{aligned} = [\begin{array}{ccc} 4 & 11 & 1 \\ â 9 & â 2 & â 7 \\ 21 & 11 & 7 \end{array}] â 3 [\begin{array}{ccc} 4 & 3 & 0 \\ â 3 & 2 & â 2 \\ 6 & 4 & 5 \end{array}] - [\begin{array}{ccc} 1 & 2 & 1 \\ 0 & 1 & â 1 \\ 3 & â 1 & 1 \end{array}] + [\begin{array}{ccc} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{array}] = O \end{aligned}$

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