Q20 / 50 · Manipal (MET) 2023 · Mathematics

If $A=\left[\begin{array}{cc}2 & -1 \\ -7 & 4\end{array}\right]$ and $B=\left[\begin{array}{ll}4 & 1 \\ 7 & 2\end{array}\right]$, then $B^T A^T$ is

  • A
    null matrix
  • Correct answer
    B
    an identity matrix
  • C
    scalar but not an identity matrix
  • D
    such that $T_r\left(B^T A^T\right)=4$

Solution

$B^T A^T=(A B)^T$ $=\left(\left[\begin{array}{cc}2 & -1 \\ -7 & 4\end{array}\right]\left[\begin{array}{cc}4 & 1 \\ 7 & 2\end{array}\right]\right)^{\top}=\left[\begin{array}{cc}8-7 & 2-2 \\ -28+28 & -7+8\end{array}\right]^{\top}$ $=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]^{\top}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=I$

Hence, the correct answer is option: 'B'.

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