Question

If \[A = \left[ {\begin{array}{*{20}{c}}4&2\\3&4\end{array}} \right]\], then find the value of \[\left| {adj A} \right|\].
A. \[16\]
B. \[10\]
C. 6
D. None of these

Answer 1

Hint: First, calculate the adjoint matrix of the given \[2 \times 2\] matrix \[A\]. Then calculate the determinant of the adjoint matrix of a matrix \[A\] to 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: \[adj A = \left[ {\begin{array}{*{20}{c}}d&{ - b}\\{ - c}&a\end{array}} \right]\]
The determinant of a \[2 \times 2\] matrix \[A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\] is: \[\left| A \right| = ad - bc\]

Complete step by step solution:
The given matrix is \[A = \left[ {\begin{array}{*{20}{c}}4&2\\3&4\end{array}} \right]\].
Let’s find out the adjoint matrix of the given matrix \[A\].
Apply the rule for the adjoint matrix of a \[2 \times 2\] matrix.
We get,
\[adj A = \left[ {\begin{array}{*{20}{c}}4&{ - 2}\\{ - 3}&4\end{array}} \right]\]

Now calculate the determinant of the above adjoint matrix.
Apply the determinant formula for a \[2 \times 2\] matrix.
We get,
\[\left| {adj A} \right| = 4 \times 4 - (\left( { - 3} \right) \times \left( { - 2} \right))\]
\[ \Rightarrow \left| {adj A} \right| = 16 - 6\]
\[ \Rightarrow \left| {adj A} \right| = 10\]
Hence the correct option is B.

Note: Students should keep in mind 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. And to find the adjoint matrix of a \[2 \times 2\] matrix: first Interchange the elements of the principal diagonal and then change the signs of the elements of the other diagonal.