If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] be such that \[A^{- 1} = k A,\] then find the value of k.

\[A = \begin{bmatrix} 2 & 3\\5 & - 2 \end{bmatrix}\]\[ \therefore \left| A \right| = \begin{vmatrix} 2 & 3\\5 & - 2 \end{vmatrix} = - 14 - 15 = - 19 \]\[\text{ The value is non - zero, so }A^{- 1} \text{ exists .} \]\[\text{ By definition, we have }\]\[ A^{- 1} A = I [\text{ I is the identity matrix}]\]\[ \Rightarrow kA . A = I [\text{ Substituting }A^{- 1} = kA]\]\[ \Rightarrow k\begin{bmatrix} 2 & 3\\5 & - 2 \end{bmatrix}\begin{bmatrix} 2 & 3\\5 & - 2 \end{bmatrix} = \begin{bmatrix} 1 & 0\\0 & 1 \end{bmatrix}\]\[ \Rightarrow k\begin{bmatrix} 4 + 15 & 6 - 6\\10 - 10 & 15 + 4 \end{bmatrix} = \begin{bmatrix} 1 & 0\\0 & 1 \end{bmatrix}\]\[ \Rightarrow k\begin{bmatrix} 19 & 0\\0 & 19 \end{bmatrix} = \begin{bmatrix} 1 & 0\\0 & 1 \end{bmatrix}\]\[ \Rightarrow k = \frac{1}{19}\]

If a = [ 2 3 5 − 2 ] Be Such that a − 1 = K a , Then Find the Value of K. - Mathematics

प्रश्न

If $A = [\begin{matrix} 2 & 3 \\ 5 & - 2 \end{matrix}]$ be such that $A^{- 1} = k A,$ then find the value of k.

उत्तर

$A = [\begin{matrix} 2 & 3 \\ 5 & - 2 \end{matrix}]$
$∴ | A | = | \begin{matrix} 2 & 3 \\ 5 & - 2 \end{matrix} | = - 14 - 15 = - 19$
$The value is non - zero, so A^{- 1} exists .$
$By definition, we have$
$A^{- 1} A = I [I is the identity matrix]$
$\Rightarrow k A . A = I [Substituting A^{- 1} = k A]$
$\Rightarrow k [\begin{matrix} 2 & 3 \\ 5 & - 2 \end{matrix}] [\begin{matrix} 2 & 3 \\ 5 & - 2 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$\Rightarrow k [\begin{matrix} 4 + 15 & 6 - 6 \\ 10 - 10 & 15 + 4 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$\Rightarrow k [\begin{matrix} 19 & 0 \\ 0 & 19 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$\Rightarrow k = \frac{1}{19}$

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

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Q 19 Q 18 Q 20

पाठ 7: Adjoint and Inverse of a Matrix - Exercise 7.3 [पृष्ठ ३५]

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पाठ 7 Adjoint and Inverse of a Matrix
Exercise 7.3 | Q 19 | पृष्ठ ३५

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If a = [ 2 3 5 − 2 ] Be Such that a − 1 = K a , Then Find the Value of K. - Mathematics

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