Using elementary transformations- find the inverse of matrix - 2 5- 1 3 - if it exists-

Name: NCERT - Grade 12 - Mathematics - Matrices - Q75
Uploaded: 2022-07-04T10:36:42+05:30
Description: Let nbsp;A= [ 2 amp; 5; nbsp; 1 amp; 3 ] We know that nbsp; A= IA ⇒ nbsp;[ 2 amp; 5; nbsp; 1 amp; 3 ] = nbsp;[ 1 amp; 0; nbsp; 0 ...

Let nbsp;A= [ 2 amp; 5; nbsp; 1 amp; 3 ] We know that nbsp; A= IA ⇒ nbsp;[ 2 amp; 5; nbsp; 1 amp; 3 ] = nbsp;[ 1 amp; 0; nbsp; 0 ...

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Byju's Answer

Standard XII

Mathematics

Elementary Transformations

Using element...

Question

Using elementary transformations, find the inverse of matrix $[\begin{matrix} 2 & 5 1 & 3 \end{matrix}]$ , if it exists.

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Solution

Let $A = [\begin{matrix} 2 & 5 1 & 3 \end{matrix}]$
We know that $A = I A$
$\Rightarrow [\begin{matrix} 2 & 5 1 & 3 \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}] A$
Applying $R_{1} \to R_{1} - R_{2}$
$\Rightarrow [\begin{matrix} 2 - 1 & 5 - 3 1 & 3 \end{matrix}] = [\begin{matrix} 1 - 0 & 0 - 1 0 & 1 \end{matrix}] A$
$\Rightarrow [\begin{matrix} 1 & 2 1 & 3 \end{matrix}] = [\begin{matrix} 1 & - 1 0 & 1 \end{matrix}] A$
Applying $R_{2} \to R_{2} - R_{1}$
$\Rightarrow [\begin{matrix} 1 & 2 1 - 1 & 3 - 2 \end{matrix}] = [\begin{matrix} 1 & - 1 0 - 1 & 1 - (- 1) \end{matrix}] A$
$\Rightarrow [\begin{matrix} 1 & 2 0 & 1 \end{matrix}] = [\begin{matrix} 1 & - 1 - 1 & 2 \end{matrix}] A$
Applying $R_{1} \to R_{1} - 2 R_{2}$
$\Rightarrow [\begin{matrix} 1 - 2 (0) & 2 - 2 (1) 0 & 1 \end{matrix}] = [\begin{matrix} 1 - 2 (- 1) & - 1 - 2 (2) - 1 & 2 \end{matrix}] A$
$\Rightarrow [\begin{matrix} 1 & 0 0 & 1 \end{matrix}] = [\begin{matrix} 3 & - 5 - 1 & 2 \end{matrix}] A$
$\Rightarrow I = [\begin{matrix} 3 & - 5 - 1 & 2 \end{matrix}] A$
This is similar to $I = A^{- 1} A$
Thus, $A^{- 1} = [\begin{matrix} 3 & - 5 - 1 & 2 \end{matrix}]$

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Standard XII Mathematics

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