Question

A=[3 -4 -1 2] ,find a matrix B such that AB=I

Answer 1

GIVEN :

The matrix $A=\left(\begin{array}{cc}3&-4\\-1&2\end{array}\right)$

TO FIND :

The matrix B such that AB=I

SOLUTION :

Given that the matrix is $A=\left(\begin{array}{cc}3&-4\\-1&2\end{array}\right)$

Now we have to find the matrix B such that AB=I

We know that the Identity Matrix for 2x2 matrix is given by

$I=\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$

Substitute the matrices in the equation AB=I we get

AB=I

$B=IA^{-1}$

The formula for $A^{-1}$ is $A^{-1}=\frac{adjA}{|A|}$

Now find |A|:

$|A|=\left|\begin{array}{cc}3&-4\\-1&2\end{array}\right|$

=6-4

⇒ |A|=2

$adjA=\left(\begin{array}{cc}2&4\\1&3\end{array}\right)$

$A^{-1}=\frac{adjA}{|A|}$

$=\frac{\left(\begin{array}{cc}2&4\\1&3\end{array}\right)}{2}$

$A^{-1}=\left(\begin{array}{cc}1&2\\\frac{1}{2}&\frac{3}{2}\end{array}\right)$

Now in $B=IA^{-1}$ we have that

$B=IA^{-1}$

$B=\left(\begin{array}{cc}1&0\\0&1\end{array}\right)\left(\begin{array}{cc}1&2\\\frac{1}{2}&\frac{3}{2}\end{array}\right)$

$B=\left(\begin{array}{cc}1&2\\\frac{1}{2}&\frac{3}{2}\end{array}\right)$

⇒ the matrix B is $\left(\begin{array}{cc}1&2\\\frac{1}{2}&\frac{3}{2}\end{array}\right)$

∴ $B=\left(\begin{array}{cc}1&2\\\frac{1}{2}&\frac{3}{2}\end{array}\right)$