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
\[\begin{align}
  & \text{det(}{{\text{A}}^{2}}\text{B}{{\text{C}}^{-1}}\text{)} \\
 & \text{=det(}{{\text{A}}^{2}}\text{)}\text{.det(B)}\text{.det(}{{\text{C}}^{-1}}\text{)} \\
 & \text{=}{{\left\{ \text{det(A)} \right\}}^{2}}.\text{det(B)}.\dfrac{1}{\det (C)} \\
\end{align}\]
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
\[\begin{align}
  & \text{=}{{\left\{ \text{det(A)} \right\}}^{2}}.\text{det(B)}.\dfrac{1}{\det (C)} \\
 & ={{3}^{2}}.3.\dfrac{1}{5} \\
 & =\dfrac{27}{5} \\
\end{align}\]
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.