Question

If A, B, and C are $n\times n$ matrices and det(A)=3,det(B)=3anddet(C)=5, then the value of the $\text{det(}{{\text{A}}^{2}}\text{B}{{\text{C}}^{-1}}\text{)}$
A.30
B.$\dfrac{27}{5}$
C.60
D.6

Answer 1

Hint: We have the determinant values of matrix A, B, and C. Using the property $\text{det}\left( \text{ABC} \right)\text{=det}\left( \text{A} \right)\text{det}\left( \text{B} \right)\text{det}\left( \text{C} \right)$ and \text{det(}{{\text{A}}^{n}}\text{)=}{{\left\\{ \text{det(A)} \right\\}}^{n}} , convert $\text{det(}{{\text{A}}^{2}}\text{B}{{\text{C}}^{-1}}\text{)}$ into a simpler form and then put the determinant values of the matrix A, B, and C.

Complete step-by-step answer:
According to the question, it is given that,
$\text{det}\left( \text{A} \right)=3$ …………….(1)
$\text{det}\left( \text{B} \right)=3$ …………….(2)
$\text{det}\left( \text{C} \right)=5$ …………….(3)
We have to find the value of $\text{det(}{{\text{A}}^{2}}\text{B}{{\text{C}}^{-1}}\text{)}$ . But we don’t have determinant values of ${{\text{A}}^{2}}$ and ${{\text{C}}^{-1}}$. We only have the determinant values of matrix A, B, and C. So, we have to transform $\text{det(}{{\text{A}}^{2}}\text{B}{{\text{C}}^{-1}}\text{)}$ into a simpler form so that the determinant values of A, B, and C can be used directly.
We know the property, $\text{det}\left( \text{ABC} \right)\text{=det}\left( \text{A} \right)\text{det}\left( \text{B} \right)\text{det}\left( \text{C} \right)$ and \text{det(}{{\text{A}}^{n}}\text{)=}{{\left\\{ \text{det(A)} \right\\}}^{n}} .
Now, using the above properties and converting $\text{det(}{{\text{A}}^{2}}\text{B}{{\text{C}}^{-1}}\text{)}$ into a simpler form, we get

& \text{det(}{{\text{A}}^{2}}\text{B}{{\text{C}}^{-1}}\text{)} \\\ & \text{=det(}{{\text{A}}^{2}}\text{)}\text{.det(B)}\text{.det(}{{\text{C}}^{-1}}\text{)} \\\ & \text{=}{{\left\\{ \text{det(A)} \right\\}}^{2}}.\text{det(B)}.\dfrac{1}{\det (C)} \\\ \end{aligned}$$ Putting the value of $$\text{det}\left( \text{A} \right)$$ , $$\text{det}\left( \text{B} \right)$$, and $$\text{det}\left( \text{C} \right)$$ from equation (1), equation (2), and equation (3), we get $$\begin{aligned} & \text{=}{{\left\\{ \text{det(A)} \right\\}}^{2}}.\text{det(B)}.\dfrac{1}{\det (C)} \\\ & ={{3}^{2}}.3.\dfrac{1}{5} \\\ & =\dfrac{27}{5} \\\ \end{aligned}$$ So, the value of $$\text{det(}{{\text{A}}^{2}}\text{B}{{\text{C}}^{-1}}\text{)}$$ is $$\dfrac{27}{5}$$ . Hence, the correct option is (B). Note:In this question, one might get confused because the determinant values of $${{\text{A}}^{2}}$$ and $${{\text{C}}^{-1}}$$ is not given in the question. So, we have to convert the determinant values of $${{\text{A}}^{2}}$$ and $${{\text{C}}^{-1}}$$ in terms of $$\text{det}\left( \text{A} \right)$$ and $$\text{det}\left( \text{C} \right)$$ . This is a reason that most students skip such questions in exams, so students must know this method of simplifying the determinants and then using the data given in the question.