====== Question 5, Exercise 2.2 ======
Solutions of Question 5 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. 

=====Question 5=====
If $X=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$ then prove that $X^{2}-4 X-5 I=0$.

** Solution. **

Given the matrix 
\begin{align}L.H.S. & =X^{2}-4 X-5 I \\
&=\begin{bmatrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{bmatrix}-4\begin{bmatrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{bmatrix}-5\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}\\
& = \begin{bmatrix}
1 + 4 + 4 & 2 + 2 + 4 & 2 + 4 + 2 \\
2 + 2 + 4 & 4 + 1 + 4 & 4 + 2 + 2 \\
2 + 4 + 2 & 4 + 2 + 2 & 4 + 4 + 1
\end{bmatrix}-\begin{bmatrix}
4 & 8 & 8 \\
8 & 4 & 8 \\
8 & 8 & 4
\end{bmatrix}- \begin{bmatrix}
5 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 5
\end{bmatrix}\\
&=\begin{bmatrix}
9 & 8 & 8 \\
8 & 9 & 8 \\
8 & 8 & 9
\end{bmatrix} - \begin{bmatrix}
4 & 8 & 8 \\
8 & 4 & 8 \\
8 & 8 & 4
\end{bmatrix} - \begin{bmatrix}
5 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 5
\end{bmatrix}\\
&= \begin{bmatrix}
5 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 5
\end{bmatrix} - \begin{bmatrix}
5 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 5
\end{bmatrix}\\
&= \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix} = O\\
&=R.H.S.\end{align}

GOOD


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