If \[A = \left[ {\begin{array}{*{20}{c}}{ - 2}&6\\{ - 5}&7\end{array}} \right]\], then find the adjoint matrix of the matrix \[A\].
A. \[\left[ {\begin{array}{*{20}{c}}7&{ - 6}\\5&{ - 2}\end{array}} \right]\]
B. \[\left[ {\begin{array}{*{20}{c}}2&{ - 6}\\5&{ - 7}\end{array}} \right]\]
C. \[\left[ {\begin{array}{*{20}{c}}7&{ - 5}\\6&{ - 2}\end{array}} \right]\]
D. None of these

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 \[A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\] is: \[adj A = \left[ {\begin{array}{*{20}{c}}d&{ - b}\\{ - c}&a\end{array}} \right]\]

Complete step by step solution:
The given matrix is \[A = \left[ {\begin{array}{*{20}{c}}{ - 2}&6\\{ - 5}&7\end{array}} \right]\].

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,
\[adj A = \left[ {\begin{array}{*{20}{c}}7&{ - (6)}\\-(-5)&{ - 2}\end{array}} \right]\]
So, \[adj A = \left[ {\begin{array}{*{20}{c}}7&{ - 6}\\5&{ - 2}\end{array}} \right]\]
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.