If A- - 2 0 -1- 5 1 0- 0 1 3 - then find A-1 using elementary row operations-

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

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Standard XII

Mathematics

Finding Inverse Using Elementary Transformations

If A= [ 2 0...

Question

If $A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$ , then find $A^{- 1}$ using elementary row operations.

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Solution

$A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$
We know that $A = I A$
$∴ ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$
Applying $R_{1} \to \frac{1}{2} R_{1},$ we have
$\Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 1 / 2 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 / 2 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$
Applying $R_{2} \to R_{2} - 5 R_{1},$ we have
$\Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 1 / 2 0 & 1 & 5 / 2 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 / 2 & 0 & 0 - 5 / 2 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$
Applying $R_{3} \to R_{3} - R_{2},$ we have
$\Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 1 / 2 0 & 1 & 5 / 2 0 & 0 & 1 / 2 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 / 2 & 0 & 0 - 5 / 2 & 1 & 0 5 / 2 & - 1 & 1 \end{matrix} ⎤ ⎥ ⎦ A$
Applying $R_{3} \to 2 R_{3},$ we have
$\Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 1 / 2 0 & 1 & 5 / 2 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 / 2 & 0 & 0 - 5 / 2 & 1 & 0 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦ A$
Applying $R_{1} \to R_{1} + \frac{1}{2} R_{3},$ and $R_{2} \to R - 2 - \frac{5}{2} R_{2},$ we have
$\Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦ A$
$\Rightarrow I = A^{- 1} A$
$∴ A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦$
Hence, the answer is $⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦ .$

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