Name: Finding Inverse by Elementary Transformations
Uploaded: 2019-07-19
Duration: 12 min 5 s
Description: In this video, we will learn to find the inverse of matrices using elementary transformations of matrices. We will learn how to find the inverse of a matrix,...

Question

Find the inverse of each of the following matrices by using elementary row transformations:

$[\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{matrix}]$

Accepted Answer

Given that $[\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{matrix}]$

We know $A = I A$ $\Rightarrow [\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{matrix}] = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] A$

$\Rightarrow [\begin{matrix} 1 & 0 & \frac{- 1}{2} \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{matrix}] = [\begin{matrix} \frac{1}{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] A [Applying R_{1} \to \frac{1}{2} R_{1}]$ $\Rightarrow [\begin{matrix} 1 & 0 & \frac{- 1}{2} \\ 0 & 1 & \frac{5}{2} \\ 0 & 1 & 3 \end{matrix}] = [\begin{matrix} \frac{1}{2} & 0 & 0 \\ \frac{- 5}{2} & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] A [Applying R_{2} \to R_{2} - 5 R_{1}]$

$\Rightarrow [\begin{matrix} 1 & 0 & \frac{- 1}{2} \\ 0 & 1 & \frac{5}{2} \\ 0 & 0 & \frac{1}{2} \end{matrix}] = [\begin{matrix} \frac{1}{2} & 0 & 0 \\ \frac{- 5}{2} & 1 & 0 \\ \frac{5}{2} & - 1 & 1 \end{matrix}] A [Applying R_{2} \to R_{2} - 5 R_{1}]$ $\Rightarrow [\begin{matrix} 1 & 0 & \frac{- 1}{2} \\ 0 & 1 & \frac{5}{2} \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} \frac{1}{2} & 0 & 0 \\ - \frac{5}{2} & 1 & 0 \\ 5 & - 2 & 2 \end{matrix}] A [Applying R_{3} \to 2 R_{3}]$

$\Rightarrow [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 3 & - 1 & 1 \\ - 15 & 6 & - 5 \\ 5 & - 2 & 2 \end{matrix}] A [Applying R_{1} \to R_{1} + \frac{1}{2} R_{3} and R_{2} \to R_{2} - \frac{5}{2} R_{3}]$

$\Rightarrow A^{- 1} = [\begin{matrix} 3 & - 1 & 1 \\ - 15 & 6 & - 5 \\ 5 & - 2 & 2 \end{matrix}]$

Find the inverse of each of the following matrices by using elementary row transformations:

$[\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{matrix}]$

Important Questions on Adjoint and Inverse of a Matrix

Learn and Prepare for any exam you want !

Find the inverse of each of the following matrices by using elementary row transformations: 20-1510013

Important Questions on Adjoint and Inverse of a Matrix

Learn and Prepare for any exam you want !

Find the inverse of each of the following matrices by using elementary row transformations:

$[\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{matrix}]$