1. Tardigrade
  2. Question
  3. Mathematics
  4. If A=[1&0&0 0&1&0 a&b&-1] and I is the unit matrix of order 3, then A2 + 2A4 + 4A6 is equal to

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Q. If $A=\begin{bmatrix}1&0&0\\ 0&1&0\\ a&b&-1\end{bmatrix}$ and I is the unit matrix of order 3, then $A^2 + 2A^4 + 4A^6$ is equal to

Matrices

Solution:

$A^{2}=\begin{bmatrix}1&0&0\\ 0&1&0\\ a&b&-1\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&1&0\\ a&b&-1\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$
$A^{2}=A^{4}=A^{6}=I_{3} \Rightarrow A^{2}+2A^{4}+4A^{6}$
$=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}+\begin{bmatrix}2&0&0\\ 0&2&0\\ 0&0&2\end{bmatrix}+\begin{bmatrix}4&0&0\\ 0&4&0\\ 0&0&4\end{bmatrix}$
$=\begin{bmatrix}7&0&0\\ 0&7&0\\ 0&0&7\end{bmatrix}=7I_{3}=7A^{8}$