If \[A = \left[ {\begin{array}{*{20}{c}}3&4\\5&7\end{array}} \right]\], then find the value of \[A\left( {adj A} \right)\].
A. \[I\]
B. \[\left| A \right|\]
C. \[\left| A \right|I\]
D. None of these

Hint: First, calculate the adjoint matrix of the given \[2 \times 2\] matrix \[A\]. Then substitute the value of \[adj A\] and \[A\] in the given expression and simplify it by using the matrix multiplication method 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: \[adj A = \left[ {\begin{array}{*{20}{c}}d&{ - b}\\{ - c}&a\end{array}} \right]\]

Complete step by step solution:
 The given matrix is \[A = \left[ {\begin{array}{*{20}{c}}3&4\\5&7\end{array}} \right]\].

Let’s calculate the adjoint matrix of the matrix \[A\].
Apply the formula of the adjoint matrix of \[2 \times 2\] matrix.
We get,
\[adj A = \left[ {\begin{array}{*{20}{c}}7&{ - 4}\\{ - 5}&3\end{array}} \right]\]

Now substitute the values of \[adj A\] and \[A\] in the given expression.
\[A\left( {adj A} \right) = \left[ {\begin{array}{*{20}{c}}3&4\\5&7\end{array}} \right]\left[ {\begin{array}{*{20}{c}}7&{ - 4}\\{ - 5}&3\end{array}} \right]\]
Apply the matrix multiplication method.
\[A\left( {adj A} \right) = \left[ {\begin{array}{*{20}{c}}{3 \times 7 + 4 \times \left( { - 5} \right)}&{3 \times \left( { - 4} \right) + 4 \times 3}\\{5 \times 7 + 7 \times \left( { - 5} \right)}&{5 \times \left( { - 4} \right) + 7 \times 3}\end{array}} \right]\]
Solve the right-hand side of the above equation.
\[A\left( {adj A} \right) = \left[ {\begin{array}{*{20}{c}}{21 - 20}&{ - 12 + 12}\\{35 - 35}&{ - 20 + 21}\end{array}} \right]\]
\[A\left( {adj A} \right) = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]\]
\[A\left( {adj A} \right) = I\] \[.....\left( 1 \right)\]

Now calculate the determinant of the given matrix \[A\].
\[\left| A \right| = 3 \times 7 - 4 \times 5\]
\[ \Rightarrow \left| A \right| = 21 - 20\]
\[ \Rightarrow \left| A \right| = 1\] \[.....\left( 2 \right)\]
We can write the equation \[\left( 1 \right)\] as follows:
\[A\left( {adj A} \right) = 1 \times I\]
From the equation \[\left( 2 \right)\], we get
\[A\left( {adj A} \right) = \left| A \right|I\]
Hence the correct options are A and C.


Note: Students should keep in mind that the product of two matrices is defined only if the number of columns of the first matrix is equal to the number of rows of the second matrix.