If A = [ * 20 c 3&1 - 1 &2 ] Show that A^2 – 5A + 7I = O. Hence find A^ -1

We have, A^2=A = 3&1 -1&2 3&1 -1&2 = 8&5 -5&2 |A| = (3)(2) - (1)(-1) = 6 + 1 = 7 0 A is non singular and hence A^ -1 exists. A^2 - 5A + 7I = [ * 20 c 8&5 - 5 &2 ] - 5 [ * 20 c 3&1 - 1 &2 ] + 7 [ * 20 c 1&0 0&1 ] = [ * 20 c 8 - 15 + 7 & 5 - 5 + 0 - 5 + 50 & 3 - 10 + 7 ] = [ * 20 c 0&0 0&0 ] = O Now, A^2 – 5A + 7I = O (given) A^2 – 5A = -7I Post multiplying by A^ -1 , we get, A^2A^ -1 -5AA^ -1 = -7IA^ -1 AAA^ -1 – 5AA^ -1 = -7IA^ -1 A – 5I = -7A^ -1 [AA^ -1 = I] 7A^ -1 = 5I – A = 5 [ * 20 c 1&0 0&1 ] - [ * 20 c 3&1 - 1 &2 ] = [ * 20 c 2&-1 1&3 ] A^ - 1 = 1 7 [ * 20 c 2& - 1 1&3 ]

If A = [ * 20 c 3&1 - 1 &2 ] Show that A^2 – 5A + 7I = O. Hence f…

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Question

If $ A = \left[ {\begin{array}{*{20}{c}} 3&1 \\ { - 1}&2 \end{array}} \right]$ Show that $A^2 – 5A + 7I = O.$ Hence find $A^{-1}$

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Answer

We have, $A^2=A$
$=\begin{bmatrix}3&1\\-1&2\end{bmatrix}\begin{bmatrix}3&1\\-1&2\end{bmatrix}=\begin{bmatrix}8&5\\-5&2\end{bmatrix}$
$|A| = (3)(2) - (1)(-1)$
$= 6 + 1$
$= 7 \neq0$
$\Rightarrow A$ is non singular and hence $A^{-1}$ exists.
$ {A^2} - 5A + 7I$
$= \left[ {\begin{array}{*{20}{c}} 8&5 \\ { - 5}&2 \end{array}} \right] - 5\left[ {\begin{array}{*{20}{c}} 3&1 \\ { - 1}&2 \end{array}} \right] + 7\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}} {8 - 15 + 7}&{5 - 5 + 0} \\ { - 5 + 50}&{3 - 10 + 7} \end{array}} \right]$
$= \left[ {\begin{array}{*{20}{c}} 0&0 \\ 0&0 \end{array}} \right]$
$= O$
Now,
$A^2 – 5A + 7I = O \ ($given$)$
$A^2 – 5A = -7I$
Post multiplying by $A^{-1},$ we get,
$A^2A^{-1}-5AA^{-1} = -7IA^{-1}$
$AAA^{-1} – 5AA^{-1} = -7IA^{-1}$
$A – 5I = -7A^{-1} [AA^{-1} = I]$
$7A^{-1} = 5I – A$
$ = 5\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 3&1 \\ { - 1}&2 \end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}} 2&-1\\ 1&3 \end{array}} \right]$
$ {A^{ - 1}} = \frac{1}{7}\left[ {\begin{array}{*{20}{c}} 2&{ - 1} \\ 1&3 \end{array}} \right]$

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