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if a bmatrix 1 0 2 0 2 1 2 0 3 bmatrix prove that
Question:
If
\(A=\begin{bmatrix}1&0&2\\ 0&2&1\\ 2&0&3\end{bmatrix}\)
, prove that
\(A^3-6A^2+7A+2I=0\)
CBSE CLASS XII
Updated On:
Feb 15, 2024
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Solution and Explanation
\(A^2=AA=\begin{bmatrix}1&0&2\\ 0&2&1\\ 2&0&3\end{bmatrix}\begin{bmatrix}1&0&2\\ 0&2&1\\ 2&0&3\end{bmatrix}\)
\(=\begin{bmatrix}1+0+4& 0+0+0& 2+0+6\\ 0+0+2& 0+4+0& 0+2+3\\ 2+0+6& 0+0+0& 4+0+9\end{bmatrix}\)
\(=\begin{bmatrix}5&0&8\\ 2&4&5\\ 8&0&13\end{bmatrix}\)
Now
\(A^3=A^2.A=\begin{bmatrix}5&0&8\\ 2&4&5\\ 8&0&13\end{bmatrix}\begin{bmatrix}1&0&2\\ 0&2&1\\ 2&0&3\end{bmatrix}\)
\(=\begin{bmatrix}5+0+16& 0+0+0& 10+0+24\\ 2+0+10& 0+8+0& 4+4+15\\ 8+0+26& 0+0+0& 16+0+39\end{bmatrix}\)
\(=\begin{bmatrix}21& 0& 34\\ 12& 8& 23\\ 34& 0& 55\end{bmatrix}\)
\(\therefore A^3-6A^2+7A+2I\)
\(=\begin{bmatrix}21& 0& 34\\ 12& 8& 23\\ 34& 0& 55\end{bmatrix}-6\begin{bmatrix}5&0&8\\ 2&4&5\\ 8&0&13\end{bmatrix}+7\begin{bmatrix}1&0&2\\ 0&2&1\\ 2&0&3\end{bmatrix}+2\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}\)
\(=\begin{bmatrix}21& 0& 34\\ 12& 8& 23\\ 34& 0& 55\end{bmatrix}-\begin{bmatrix}30& 0& 48\\ 12& 24& 30\\ 48& 0& 78\end{bmatrix}+\begin{bmatrix}7& 0& 14\\ 0& 14& 7\\ 14& 0& 21 \end{bmatrix}+\begin{bmatrix}2&0&0\\ 0&2&0\\ 0&0&2\end{bmatrix}\)
\(=\begin{bmatrix}30& 0& 48\\ 12& 24& 30\\ 48& 0& 78\end{bmatrix}-\begin{bmatrix}30& 0& 48\\ 12& 24& 30\\ 48& 0& 78\end{bmatrix}\)
\(=\begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix}=0\)
\(\therefore A^3-6A^2+7A+2I=0\)
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