Question

If $$A=\left[\begin{array}{cc}-2& 6\\ -5& 7\end{array}\right]$$, then find the adjoint matrix of the matrix $$A$$.
A. $$\left[\begin{array}{cc}7& -6\\ 5& -2\end{array}\right]$$
B. $$\left[\begin{array}{cc}2& -6\\ 5& -7\end{array}\right]$$
C. $$\left[\begin{array}{cc}7& -5\\ 6& -2\end{array}\right]$$
D. None of these

Answer 1

Hint: Here, a $$2\times 2$$ matrix is given. Use the rule to calculate the adjoint matrix of a $$2\times 2$$ matrix and get the required answer.

Formula used:
The adjoint matrix of a $$2\times 2$$ matrix $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$ is: $$adjA=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$

Complete step by step solution:
The given matrix is $$A=\left[\begin{array}{cc}-2& 6\\ -5& 7\end{array}\right]$$.

Let’s find out the adjoint matrix of the given matrix $$A$$.
Apply the rule to calculate the adjoint matrix of a $$2\times 2$$ matrix.
Here we have, $$a=-2$$
$$b=6$$
$$c=-5$$
$$d=7$$
We get,
$$adjA=\left[\begin{array}{cc}7& -(6)\\ -(-5)& -2\end{array}\right]$$
So, $$adjA=\left[\begin{array}{cc}7& -6\\ 5& -2\end{array}\right]$$
Hence the correct option is A.

Note: We know that the adjoint matrix of any matrix is the transpose of its cofactor matrix. But for a $$2\times 2$$ matrix, we don’t need to calculate the cofactor matrix. To find the adjoint matrix of a $$2\times 2$$ matrix students should remember the following steps:
Interchange the elements of the principal diagonal.
Change the signs of the elements of the other diagonal.