Question

Find the adjoint and the inverse of the matrix $\left[\begin{array}{ccc}2&-3\\4&6\end{array}\right]$

Answer 1

Answer:

$adj.A=\left[\begin{array}{ccc}2&-3\\4&6\end{array}\right]$

$A^{-1}=\left[\begin{array}{ccc}\frac{1}{4}&\frac{1}{8}\\\\\frac{-1}{6}&\frac{1}{12}\end{array}\right]$

Step-by-step explanation:

As we know that Adjoint of matrix is calculated as Minor × Co-factor of each element and taking transpose of it.

or

$adj.A=[A_{ji}]_{n\times n}$$A^{-1}$

$A= \left[\begin{array}{ccc}2&-3\\4&6\end{array}\right]\\\\adj.A=\left[\begin{array}{ccc}6&3\\-4&2\end{array}\right]^{'}\\\\adj.A=\left[\begin{array}{ccc}2&-3\\4&6\end{array}\right]$

Now

$A^{-1} =\frac{adj.A}{|A|} \\\\|A|=24\\\\\\A^{-1} =\frac{1}{24} \left[\begin{array}{ccc}2&-3\\4&6\end{array}\right]$

$A^{-1}=\left[\begin{array}{ccc}\frac{1}{4}&\frac{1}{8}\\\\\frac{-1}{6}&\frac{1}{12}\end{array}\right]$