Name: Orthogonal Matrices
Uploaded: 2019-07-19
Duration: 4 min 43 s
Description: In this video, we will learn the concept of dot products and the applications of matrices and their orthogonality and orthonormality. Let us learn more.

Question

By using elementary operation , find the inverse of the matrix

A = [\begin{array}{r} 1 & 2 \\ 2 & - 1 \end{array}]

Accepted Answer

Given that :- $A = [\begin{matrix} 1 & 2 \\ 2 & - 1 \end{matrix}]$

In order to use elementary row operation we may write $A = IA$

or $[\begin{matrix} 1 & 2 \\ 2 & - 1 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] A$

Then, $[\begin{matrix} 1 & 2 \\ 0 & - 5 \end{matrix}] = [\begin{matrix} 1 & 0 \\ - 2 & 1 \end{matrix}] A$ (applying $R_{2} \to R_{2} - 2 R_{1}$ )

$[\begin{array}{l} 1 & 2 \\ 0 & 1 \end{array}] = [\begin{matrix} 1 & 0 \\ \frac{2}{5} & \frac{- 1}{5} \end{matrix}] A$ (applying $R_{2} \to - \frac{1}{5} R_{2}$ )

$[\begin{array}{l} 1 & 2 \\ 0 & 1 \end{array}] = [\begin{matrix} 1 & 0 \\ \frac{2}{5} & \frac{- 1}{5} \end{matrix}] A$ (applying $R_{1} \to R_{1} - 2 R_{2}$ )

Thus

$A^{- 1} = [\begin{array}{ll} \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{- 1}{5} \end{array}]$

Alternatively ,in order to use elementary column operation we write $A = AI$ i.e

applying $C_{1} \to C_{2} - 2 C_{1}$ ,we get

$[\begin{array}{cc} 1 & 0 \\ 2 & - 5 \end{array}] = A [\begin{array}{cc} 1 & - 2 \\ 0 & 1 \end{array}]$

Now applying $C_{2} \to - \frac{1}{5} C_{2}$

$[\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}] = A [\begin{array}{ll} 1 & \frac{2}{5} \\ 0 & \frac{- 1}{5} \end{array}]$

Now , applying $C_{1} \to C_{1} - 2 C_{2}$ we obtain

$[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}] = A [\begin{array}{ll} \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{- 1}{5} \end{array}]$

Hence $A^{- 1} = [\begin{array}{cc} \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{- 1}{5} \end{array}]$

By using elementary operation , find the inverse of the matrix $A = [\begin{array}{r} 1 & 2 \\ 2 & - 1 \end{array}]$

Important Questions on Matrices

Learn and Prepare for any exam you want !

By using elementary operation , find the inverse of the matrix A=122-1

Important Questions on Matrices

Learn and Prepare for any exam you want !

By using elementary operation , find the inverse of the matrix $A = [\begin{array}{r} 1 & 2 \\ 2 & - 1 \end{array}]$