If \[A = \begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix}\] , verify that \[A^2 - 4 A + I = O,\text{ where }I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\text{ and }O = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\] . Hence, find A−1.

\[A = \begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix}\]\[ \therefore A^2 = \begin{bmatrix}7 & 12 \\ 4 & 7\end{bmatrix}\]and\[ A^2 - 4A + I = \begin{bmatrix}7 & 12 \\ 4 & 7\end{bmatrix} - \begin{bmatrix}8 & 12 \\ 4 & 8\end{bmatrix} + \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]\[ \Rightarrow A^2 - 4A + I = \begin{bmatrix}7 - 8 + 1 & 12 - 12 + 0 \\ 4 - 4 + 0 & 7 - 8 + 1\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} = O\]\[ \Rightarrow A^2 - 4A + I = 0\]\[ \Rightarrow A^2 - 4A = - I\]\[ \Rightarrow A^{- 1} A^2 - 4A A^{- 1} = - I A^{- 1} \left[\text{ Pre - multiplying both sides by }A^{- 1} \right]\]\[ \Rightarrow A - 4I = - A^{- 1} \]\[ \Rightarrow A^{- 1} = 4I - A\]\[ \Rightarrow A^{- 1} = \left\{ \begin{bmatrix}4 & 0 \\ 0 & 4\end{bmatrix} - \begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix} \right\} = \begin{bmatrix}2 & - 3 \\ - 1 & 2\end{bmatrix}\]

If a = [ 2 3 1 2 ] , Verify that a 2 − 4 a + I = O , Where I = [ 1 0 0 1 ] and O = [ 0 0 0 0 ] . Hence, Find A−1. - Mathematics

प्रश्न

If $A = [\begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix}]$ , verify that $A^{2} - 4 A + I = O, where I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] and O = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ . Hence, find A⁻¹.

उत्तर

$A = [\begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix}]$
$∴ A^{2} = [\begin{matrix} 7 & 12 \\ 4 & 7 \end{matrix}]$
and
$A^{2} - 4 A + I = [\begin{matrix} 7 & 12 \\ 4 & 7 \end{matrix}] - [\begin{matrix} 8 & 12 \\ 4 & 8 \end{matrix}] + [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$\Rightarrow A^{2} - 4 A + I = [\begin{matrix} 7 - 8 + 1 & 12 - 12 + 0 \\ 4 - 4 + 0 & 7 - 8 + 1 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] = O$
$\Rightarrow A^{2} - 4 A + I = 0$
$\Rightarrow A^{2} - 4 A = - I$
$\Rightarrow A^{- 1} A^{2} - 4 A A^{- 1} = - I A^{- 1} [Pre - multiplying both sides by A^{- 1}]$
$\Rightarrow A - 4 I = - A^{- 1}$
$\Rightarrow A^{- 1} = 4 I - A$
$\Rightarrow A^{- 1} = {[\begin{matrix} 4 & 0 \\ 0 & 4 \end{matrix}] - [\begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix}]} = [\begin{matrix} 2 & - 3 \\ - 1 & 2 \end{matrix}]$

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Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method

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Q 17 Q 16 Q 18

अध्याय 7: Adjoint and Inverse of a Matrix - Exercise 7.1 [पृष्ठ २३]

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अध्याय 7 Adjoint and Inverse of a Matrix
Exercise 7.1 | Q 17 | पृष्ठ २३

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If a = [ 2 3 1 2 ] , Verify that a 2 − 4 a + I = O , Where I = [ 1 0 0 1 ] and O = [ 0 0 0 0 ] . Hence, Find A−1. - Mathematics

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If a = [ 2 3 1 2 ] , Verify that a 2 − 4 a + I = O , Where I = [ 1 0 0 1 ] and O = [ 0 0 0 0 ] . Hence, Find A−1. - Mathematics

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