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Question
If \[A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2\end{bmatrix},\text{ then aI + bA + 2 }A^2\] equals ____________ .
Options
A
-A
ab A
none of these
Solution
None of these
\[A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 + a & b & 3 \\ a & b & 2 \\ 3a & 2b & a + b + 4\end{bmatrix}\]
Now,
\[aI + bA + 2 A^2 = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix} + \begin{bmatrix}b & 0 & b \\ 0 & 0 & b \\ ab & b^2 & 2b\end{bmatrix} + \begin{bmatrix}2 + 2a & 2b & 6 \\ 2a & 2b & 4 \\ 6a & 6b & 2a + 2b + 8\end{bmatrix}\]
\[ = \begin{bmatrix}3a + b + 2 & 2b & b + 6 \\ 2a & a + 2b & b + 4 \\ ab + 6a & b^2 + 6b & 3a + 4b + 8\end{bmatrix}\]
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