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

A=[ 2 amp;0 amp; 1; 5 amp;1 amp;0; 0 amp;1 amp;3 ] A=IA [ 2 amp;0 amp; 1; 5 amp;1 amp;0; 0 amp;1 amp;3 ]=[ 1 amp;0 a ...

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Standard XII

Mathematics

Evaluation of a Determinant

If A =[ 2 0 -...

Question

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

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Solution

$A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$
$A = I A$
$⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$
$R_{1} \to \frac{1}{2} R_{1}$
$⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & \frac{- 1}{2} 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦ A$
$R_{2} \to R_{2} - 5 R_{1}$
$⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & \frac{- 1}{2} 0 & 1 & \frac{5}{2} 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} & 0 & 0 \frac{- 5}{2} & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ A$
$R_{3} \to R_{3} - R_{2}$
$⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & \frac{- 1}{2} 0 & 1 & \frac{5}{2} 0 & 0 & \frac{1}{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} & 0 & 0 \frac{- 5}{2} & 1 & 0 \frac{5}{2} & - 1 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ A$
$R_{3} \to 2 R_{3}$
$⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & \frac{- 1}{2} 0 & 1 & \frac{5}{2} 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} & 0 & 0 \frac{- 5}{2} & 1 & 0 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ A$
$R_{2} \to R_{2} - \frac{5}{2} R_{3}$
$⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & \frac{- 1}{2} 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} & 0 & 0 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦ A$
$R_{1} \to R_{1} + \frac{1}{2} R_{3}$
$⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦ A$
$\Rightarrow A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦$

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