Question

Find the inverse of the matrix ( if exists ) and state the reason if it doesn't exists.

$\left[ \begin{array} {ccc}2&3&4 \\ 4&3&1 \\ 1&2&4\end{array}\right]$




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Answer 1

Step-by-step explanation:

Given:$\left[ \begin{array} {ccc}2&3&4 \\ 4&3&1 \\ 1&2&4\end{array}\right]$

To find: Inverse of matrix (if exists)

Solution:Inverse of a matrix exists if given matrix is non-singular.

Step 1: Find the determinant of given matrix

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

|A|=2(12-2)-3(16-1)+4(8-3)

|A|=2×10-3×15+4×5

|A|=20-45+20

|A|=-5

|A|≠0

Thus,

Inverse of matrix exist.

Step 2: Write matrix of Minors and co-factor matrix

$M=\left[ \begin{array} {ccc}10&15&5 \\ 4&4&1 \\ -9&-14&-6\end{array}\right]$

$C=\left[ \begin{array} {ccc}10&-15&5 \\ -4&4&-1 \\ -9&14&-6\end{array}\right]$

Step 3:Write adjoint matrix by writing transpose of C.

$A_{adj}=\left[ \begin{array} {ccc}10&-4&-9 \\ -15&4&14 \\ 5&-1&-6\end{array}\right]$

Step 4: Write inverse of A

$A^{-1}=\frac{1}{|A|}A_{adj}$

$A^{-1}=\frac{-1}{5}\left[ \begin{array} {ccc}10&-4&-9 \\ -15&4&14 \\ 5&-1&-6\end{array}\right]$

Final answer:

Inverse of matrix is

$\frac{-1}{5}\left[ \begin{array} {ccc}10&-4&-9 \\ -15&4&14 \\ 5&-1&-6\end{array}\right]$

Hope it helps you.

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