Question

Find the inverse matrix $\left(A^{-1}\right)$ of the matrix $A=\left|\begin{array}{ccc}1& 2& 3\\ 2& 4& 5\\ 3& 5& 6\end{array}\right|$

Answer 1

Given matrix:

$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 5\\ 3& 5& 6\end{array}\right]$

Determinant value:

$\Delta =1(24-25)-2(12-15)+3(10-12)=-1+6-6=-1$

Thus matrix is non singular and inverse exists.

Cofactor matrix:

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

Adjoint matrix is transpose of cofactor matrix:

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

Inverse of matrix = adjoint divided by determinant value:

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