If \[A = \begin{bmatrix}3 & - 5 \\ - 4 & 2\end{bmatrix}\] , find A2 − 5A − 14I.

\[Given: A = \begin{bmatrix}3 & - 5 \\ - 4 & 2\end{bmatrix}\] \[Now, \] \[ A^2 = AA\] \[ \Rightarrow A^2 = \begin{bmatrix}3 & - 5 \\ - 4 & 2\end{bmatrix}\begin{bmatrix}3 & - 5 \\ - 4 & 2\end{bmatrix}\] \[ \Rightarrow A^2 = \begin{bmatrix}9 + 20 & - 15 - 10 \\ - 12 - 8 & 20 + 4\end{bmatrix}\] \[ \Rightarrow A^2 = \begin{bmatrix}29 & - 25 \\ - 20 & 24\end{bmatrix}\] \[\] \[ A^2 - 5A - 14I\] \[ \Rightarrow A^2 - 5A - 14I = \begin{bmatrix}29 & - 25 \\ - 20 & 24\end{bmatrix} - 5\begin{bmatrix}3 & - 5 \\ - 4 & 2\end{bmatrix} - 14\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] \[ \Rightarrow A^2 - 5A - 14I = \begin{bmatrix}29 & - 25 \\ - 20 & 24\end{bmatrix} - \begin{bmatrix}15 & - 25 \\ - 20 & 10\end{bmatrix} - \begin{bmatrix}14 & 0 \\ 0 & 14\end{bmatrix}\] \[ \Rightarrow A^2 - 5A - 14I = \begin{bmatrix}29 - 15 - 14 & - 25 + 25 + 0 \\ - 20 + 20 + 0 & 24 - 10 - 14\end{bmatrix}\] \[ \Rightarrow A^2 - 5A - 14I = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]

If A= `[[3 -5],[-4 2]]` , Find A2 − 5a − 14i. - Mathematics

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If $A = [\begin{matrix} 3 & - 5 \\ - 4 & 2 \end{matrix}]$ , find A² − 5A − 14I.

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Solution

$G i v e n : A = [\begin{matrix} 3 & - 5 \\ - 4 & 2 \end{matrix}]$
$N o w,$
$A^{2} = A A$
$\Rightarrow A^{2} = [\begin{matrix} 3 & - 5 \\ - 4 & 2 \end{matrix}] [\begin{matrix} 3 & - 5 \\ - 4 & 2 \end{matrix}]$
$\Rightarrow A^{2} = [\begin{matrix} 9 + 20 & - 15 - 10 \\ - 12 - 8 & 20 + 4 \end{matrix}]$
$\Rightarrow A^{2} = [\begin{matrix} 29 & - 25 \\ - 20 & 24 \end{matrix}]$

$A^{2} - 5 A - 14 I$
$\Rightarrow A^{2} - 5 A - 14 I = [\begin{matrix} 29 & - 25 \\ - 20 & 24 \end{matrix}] - 5 [\begin{matrix} 3 & - 5 \\ - 4 & 2 \end{matrix}] - 14 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$\Rightarrow A^{2} - 5 A - 14 I = [\begin{matrix} 29 & - 25 \\ - 20 & 24 \end{matrix}] - [\begin{matrix} 15 & - 25 \\ - 20 & 10 \end{matrix}] - [\begin{matrix} 14 & 0 \\ 0 & 14 \end{matrix}]$
$\Rightarrow A^{2} - 5 A - 14 I = [\begin{matrix} 29 - 15 - 14 & - 25 + 25 + 0 \\ - 20 + 20 + 0 & 24 - 10 - 14 \end{matrix}]$
$\Rightarrow A^{2} - 5 A - 14 I = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

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Algebraic Operations on Matrices - Multiplication of Matrices

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Chapter 5: Algebra of Matrices - Exercise 5.3 [Page 43]

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Exercise 5.3 | Q 33 | Page 43

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If A= $[\begin{matrix} 3 - 5 \\ - 42 \end{matrix}]$ , Find A2 − 5a − 14i. - Mathematics

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