Question

If $\text{A}\left(\text{adjA}\right)=5\text{I}$ where $\text{I}$ is the identity matrix of order 3, then $|adjA|$ is equal to
A. 125
B. 25
C. 5
D. 10

Answer 1

Hint: Use property of inverse of A and determinant of adjoint of A. Also two matrices are equal to each other then, the order of both the matrices will be equal.

Given, $\text{A}\left(\text{adjA}\right)=5\text{I}$ where order of identity matrix is 3.
Clearly, the order of matrix A and that of identity matrix are equal.
So, the order of matrix A is also 3.
As we know that inverse of any matrix A is given by ${\text{A}}^{-1}={\displaystyle \frac{1}{|A|}}\left(\text{adjA}\right)$ where |A| is the determinant of matrix A and adjA is the adjoint matrix of matrix A.
$\therefore \text{ A}\left[{\text{A}}^{-1}\right]=\text{A}\left[{\displaystyle \frac{1}{|A|}}\left(\text{adjA}\right)\right]={\displaystyle \frac{\text{A}\left(\text{adjA}\right)}{|A|}}={\displaystyle \frac{5\text{I}}{|A|}}$
Also, we know that $\text{ A}\left[{\text{A}}^{-1}\right]=\text{I}$ where $\text{I}$ is the identity matrix order 3
Therefore, $$\begin{gathered} \Rightarrow \text{I}={\displaystyle \frac{5\text{I}}{|A|}} \\ \Rightarrow |\text{A}|I=5I \end{gathered}$$
On comparing the above equation, we get
Determinant of the matrix A, $|\text{A}|=5$
Using the identity, $$|\text{adjA}|={\left[|A|\right]}^{n-1}$$ where n is the order of the matrix of A
Put $|\text{A}|=5$ and $\text{n}=3$ in the above identity, we have
$$\Rightarrow |\text{adjA}|={\left[5\right]}^{3-1}=5^2=25$$
Therefore, the determinant of matrix adjA is 25.
Option B is correct.

Note- Here, the inverse matrix only exists for non-singular matrices (i.e., determinant of that matrix whose inverse is required should always be non-zero). Also if in an equation two matrices are equal to each other then, order of both the matrices will be equal.