Question

The adjoint of the matrix $\left[\begin{array}{ccc}1& -2& 3\\ 0& 2& -1\\ -4& 5& 2\end{array}\right]$ is:

• A. $\left[\begin{array}{ccc}9& 19& -4\\ 4& 14& 1\\ 8& 3& 2\end{array}\right]$
• B. $\left[\begin{array}{ccc}9& 4& 8\\ 19& 14& 3\\ -4& 1& 2\end{array}\right]$
• C. $\left[\begin{array}{ccc}9& -19& -4\\ -4& 14& -1\\ 8& -3& 2\end{array}\right]$
• D. none of these

Answer 1

The correct option is A $\left[\begin{array}{ccc}9& 19& -4\\ 4& 14& 1\\ 8& 3& 2\end{array}\right]$

Cofactor of the matrix is given by

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

where$M11$= $\left[\begin{array}{cc}2& -1\\ 5& 2\end{array}\right]=4+5=9$

$M12$ = $\left[\begin{array}{cc}0& -1\\ -4& 2\end{array}\right]=0-4=-4$

$M13$ = $\left[\begin{array}{cc}0& 2\\ -4& 5\end{array}\right]=0+8=8$

$M21$ = $\left[\begin{array}{cc}-2& 3\\ 5& 2\end{array}\right]=-4-15=-19$

$M22$= $\left[\begin{array}{cc}1& 3\\ -4& 2\end{array}\right]=2+12=14$

$M23$ = $\left[\begin{array}{cc}1& -2\\ -4& 5\end{array}\right]=5-8=-3$

$M31$ = $\left[\begin{array}{cc}-2& 3\\ 2& -1\end{array}\right]=2-6=-4$

$M32$ = $\left[\begin{array}{cc}1& 3\\ 0& -1\end{array}\right]=-1-0=-1$

$M33$= $\left[\begin{array}{cc}1& -2\\ 0& 2\end{array}\right]=2-0=2$

substituting all the values in the Cofactor of the matrix formula we get,

= $\left[\begin{array}{ccc}9& 4& 8\\ 19& 14& 3\\ -4& 1& 2\end{array}\right]$

the adjoint of the matrix is transpose of cofactor matrix

$\therefore$ Adjoint of the matrix is

$\left[\begin{array}{ccc}9& 19& -4\\ 4& 14& 1\\ 8& 3& 2\end{array}\right]$