Question

The inverse of the matrix $\left\{\begin{array}{ccc}7& -3& -3\\ -1& 1& 0\\ -1& 0& 1\end{array}\right\}=$

• A. $\left\{\begin{array}{ccc}1& 1& 1\\ 3& 4& 3\\ 3& 3& 4\end{array}\right\}$
• B. $\left\{\begin{array}{ccc}1& 3& 1\\ 4& 3& 8\\ 3& 4& 1\end{array}\right\}$
• C. $\left\{\begin{array}{ccc}1& 1& 1\\ 3& 3& 4\\ 3& 4& 3\end{array}\right\}$
• D. $\left\{\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right\}$

Answer 1

The correct option is D $\left\{\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right\}$

The inverse of matrix A is =

$A^{-1}=\frac{adj\left(A\right)}{Det\left(A\right)}$

The determinant of the matrix is =

$=7\left(1\right)+3(-1)-3\left(1\right)$

$=1$

And the cofactor of the matrix is

$=\left[\begin{array}{ccc}M11& -M12& M13\\ -M21& M22& -M23\\ M31& -M32& M33\end{array}\right]$

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

$\therefore$ Adjoint of the matrix is transpose of cofactor matrix

$i.e.$ adjoint of the matrix is

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

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

Hence the answer is option D