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 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

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.

Open in App

Solution

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

Suggest Corrections

Similar questions

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦

, then find

A^{- 1}

using elementary row operations.

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦

, the find

A^{- 1}

using elementary row operations.

Q. If

A = [\begin{matrix} 1 & - 2 2 & - 6 \end{matrix}]

, find

A^{- 1}

using elementary row operations only.

Using elementary transformations, find the inverse of matrix $⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$ , if it exists.

Q. In applying one or more row operations while finding A^–1 by elementary row operations, we obtain all zeroes in one or more row, then A^–1 __________.

Join BYJU'S Learning Program

If A=⎡⎢⎣20−1510013⎤⎥⎦, the find A−1 using elementary row operations.

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