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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Matrices

NDA 2021 · Mathematics · Matrices

If A and B are square matrices of order 2 such that (A B)= (B A) ,...

If AA and BB are square matrices of order 2 such that det⁡⁡(AB)=det⁡⁡(BA)\operatorname\det(A B)=\operatorname\det(B A), then which one of the following is correct?

Options

  1. AAA must be a unit matrix.
  2. BBB must be a unit matrix.
  3. CBoth AA and BB must be unit matrices.
  4. DAA and BB need not be unit matrices.

Correct answer

D. AA and BB need not be unit matrices.

Step-by-step solution

Let A=\left|2002\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right|_{2 \times 2} and B=\left|1004\begin{array}{ll}1 & 0 \\ 0 & 4\end{array}\right|_{2 \times 2}∣A∣=4 and ∣B∣=4.det⁡⁡(A⋅B)=∣2002∣⋅∣1004∣=∣2008∣=16\begin{aligned} & |A|=4 \text{ and }|B|=4. \\ & \operatorname\det(A \cdot B)=\left|\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right| \cdot\left|\begin{array}{ll}1 & 0 \\ 0 & 4\end{array}\right|=\left|\begin{array}{ll}2 & 0 \\ 0 & 8\end{array}\right|=16\end{aligned}\operatorname\det(B \cdot A)=\left|1004\begin{array}{ll}1 & 0 \\ 0 & 4\end{array}\right| \cdot\left|2002\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right|=\left|2008\begin{array}{ll}2 & 0 \\ 0 & 8\end{array}\right|=16 Here det⁡⁡(A⋅B)=det⁡⁡(B⋅A)\operatorname\det(A \cdot B)=\operatorname\det(B \cdot A), but AandandBarenotunitmatrices.are not unit matrices.
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