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 the matrices given below.

$[\begin{array}{l} a & b \\ c & d \end{array}]$ , when $(a b - b c) \neq 0$

Accepted Answer

Here, $A = [\begin{array}{l} a & b \\ c & d \end{array}]$

We have to find $A^{- 1}$ and $A^{- 1} = \frac{a d j A}{| A |}$

Firstly, we find the adjoint $A$ and for that we have to find co-factors:

$a_{11}$ (co – factor of $a$ ) $= (- 1)^{1 + 1} (d) = (- 1)^{2} (d) = d$

$a_{12}$ (co – factor of $b$ ) $= (- 1)^{1 + 2} (c) = (- 1)^{3} (c) = - c$

$a_{21}$ (co – factor of $C$ ) $(a b - b c) \neq 0$

$a_{22}$ (co – factor of $d$ ) $= (- 1)^{2 + 2} (a) = (- 1)^{4} (a) = a$

The co-factor matrix $= [\begin{matrix} d & - c \\ - b & a \end{matrix}]$

Now, $a d j A =$ Transpose of co-factor Matrix

$∴ adj A = {[\begin{matrix} d & - c \\ - b & a \end{matrix}]}^{T} = [\begin{matrix} d & - b \\ - c & a \end{matrix}]$

Calculating $| A |$

$| A | = |\begin{array}{l} a & b \\ c & d \end{array}|$

If $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ then determinant of $A$ , is given by

$| A | = |\begin{array}{l} a & b \\ c & d \end{array}| = a d - b c$

$= [a \times d - c \times b]$

$= a d - b c$

$∴ A^{- 1} = \frac{a d j A}{| A |} = \frac{[\begin{matrix} d & - b \\ - c & a \end{matrix}]}{(a d - b c)}$

$= \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$

Hence, $A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$

Find the inverse of the matrices given below.

$[\begin{array}{l} a & b \\ c & d \end{array}]$ , when $(a b - b c) \neq 0$

Important Questions on Adjoint And Inverse Of A Matrix

Learn and Prepare for any exam you want !

Find the inverse of the matrices given below. abcd, when (ab–bc)≠0

Important Questions on Adjoint And Inverse Of A Matrix

Learn and Prepare for any exam you want !

Find the inverse of the matrices given below.

$[\begin{array}{l} a & b \\ c & d \end{array}]$ , when $(a b - b c) \neq 0$