Question

If we are given the following data about a matrix A, that is, ${{A}_{3\times 3}}$ and $\det A=5$, then $\det \left( adjA \right)=$
A. 5
B. 25
C. 125
D. $\dfrac{1}{5}$

Answer 1

Hint: We know that if we have a matrix A of order n and its determinant value is equal to ‘k’ then the determinant of adjoint of A is given as follows:
$\det \left( adjA \right)={{\left| A \right|}^{n-1}}={{k}^{n-1}}$

Complete step-by-step answer:

In this question, we have been given a matrix A of the order $\left( 3\times 3 \right)$ and det A = 5.
Now, we know that ${{A}^{-1}}=\dfrac{adjA}{\left| A \right|}$.
Also, we know that $A.{{A}^{-1}}=I$.
By substituting the value of ${{A}^{-1}}$ in the above expression, we will get as follows:

& A\left( \dfrac{adjA}{\left| A \right|} \right)=I \\\ & \Rightarrow \left| A.adj\left( A \right) \right|=\left| A \right|.I \\\ & \Rightarrow \left| A \right|.\left| adj\left( A \right) \right|={{\left| A \right|}^{n}} \\\ & \Rightarrow \left| adj\left( A \right) \right|=\dfrac{{{\left| A \right|}^{n}}}{\left| A \right|} \\\ & \Rightarrow \left| adj\left( A \right) \right|={{\left| A \right|}^{n-1}} \\\ \end{aligned}$$ Hence we can see that if we have a matrix A of order n and its determinant is equal to $$\left| A \right|$$ then the determinant of adjoint of A i.e. adjA is equal to $${{\left| A \right|}^{n-1}}$$. Here, we have n = 3 and $$\left| A \right|=5$$ $$\Rightarrow \det \left[ adj\left( A \right) \right]={{\left| A \right|}^{n-1}}={{\left( 5 \right)}^{3-1}}={{5}^{2}}=25$$ Therefore the correct answer of the question is option B. Note: Remember that the adjoint of any matrix A is the transpose of the cofactor matrix of A and it is denoted by “adj(A)”. It is also known as an adjugate matrix. Also, remember the formula of determinant of adjoint of matrix A since it will take your time to find the expression so it is better to memorize the formula which is $$\det \left( adjA \right)={{\left| A \right|}^{n-1}}={{k}^{n-1}}$$