Question

find the inverse of matrix A by using adjoint method where a is equals to

(1 0 1)
A= (0 2 3)
(1 2 1)​

Answer 1

Step-by-step explanation:

Given:

$A=\left[\begin{array}{ccc}1&0&1\\0&2&3\\1&2&1\end{array}\right]$

To find:Inverse of A using adjoint method.

Solution:

Step 1: Write formula for A-1

$A^{-1}=\frac{Adj(A)}{|A|}$

Step 2: To find adjoint we first write co-factor matrix

$C_{11}=\left|\begin{array}{cc}2&3\\2&1\end{array}\right|$

$C_{11}=2-6\\C_{11}=-4$

$C_{12}=-\left|\begin{array}{cc}0&3\\1&1\end{array}\right|$

$C_{12}=-(0-3)\\C_{12}=3$

$C_{13}=\left|\begin{array}{cc}0&2\\1&2\end{array}\right|$

$C_{13}=0-2\\C_{13}=-2$

like these three elements of co-factor matrix,find remaining elements also,just writing determinant for that element by hiding that row and column.

Thus,

$C=\left[\begin{array}{ccc}-4&3&-2\\2&0&-2\\-2&-3&2\end{array}\right]$

$Adj(A)=C^T$

$Adj(A)=\left[\begin{array}{ccc}-4&2&-2\\3&0&-3\\-2&-2&2\end{array}\right]$

Step 3: Find determinant as per formula

|A|=1(-4)-0(3)+1(-2)

|A|=-4-2

|A|=-6

Step 4: Write A-1 using determinant and adjoint

$A^{-1}=\frac{-1}{6}\left[\begin{array}{ccc}-4&2&-2\\3&0&-3\\-2&-2&2\end{array}\right]$

or

$A^{-1}=\frac{1}{6}\left[\begin{array}{ccc}4&-2&2\\-3&0&3\\2&2&-2\end{array}\right]$

$A^{-1}=\frac{1}{6}\left[\begin{array}{ccc}4&-2&2\\-3&0&3\\2&2&-2\end{array}\right]$

or

$A^{-1}=\left[\begin{array}{ccc}\frac{4}{6}&-\frac{2}{6}&\frac{2}{6}\\\\-\frac{3}{6}&0&\frac{3}{6}\\\\\frac{2}{6}&\frac{2}{6}&-\frac{2}{6}\end{array}\right]$

or

$A^{-1}=\left[\begin{array}{ccc}\frac{2}{3}&-\frac{1}{3}&\frac{1}{3}\\\\-\frac{1}{2}&0&\frac{1}{2}\\\\\frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\end{array}\right]$

Final answer:

$A^{-1}=\left[\begin{array}{ccc}\frac{2}{3}&-\frac{1}{3}&\frac{1}{3}\\\\-\frac{1}{2}&0&\frac{1}{2}\\\\\frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\end{array}\right]$

Hope it helps you.

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