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Mathematics Question on Product of Matrices

 If P=[−1 2 1] and Q=[2−41] are two matrices, then (PQ)T will be:\text{ If } P = \begin{bmatrix} -1 \\\ 2 \\\ 1 \end{bmatrix} \text{ and } Q = \begin{bmatrix} 2 & -4 & 1 \end{bmatrix} \text{ are two matrices, then } (PQ)^T \text{ will be:}

A

[457 −3−30 0−3−2]\begin{bmatrix} 4 & 5 & 7 \\\ -3 & -3 & 0 \\\ 0 & -3 & -2 \end{bmatrix}

B

[−242 4−8−4 −121]\begin{bmatrix} -2 & 4 & 2 \\\ 4 & -8 & -4 \\\ -1 & 2 & 1 \end{bmatrix}

C

[552 767 −9−70]\begin{bmatrix} 5 & 5 & 2 \\\ 7 & 6 & 7 \\\ -9 & -7 & 0 \end{bmatrix}

D

[−248 757 −8−26]\begin{bmatrix} -2 & 4 & 8 \\\ 7 & 5 & 7 \\\ -8 & -2 & 6 \end{bmatrix}

Answer

[−242 4−8−4 −121]\begin{bmatrix} -2 & 4 & 2 \\\ 4 & -8 & -4 \\\ -1 & 2 & 1 \end{bmatrix}

Explanation

Solution

To find (PQ)′(PQ)', we first compute the product PQPQ where:

  • PP is a column matrix of order 3×13 \times 1
  • QQ is a row matrix of order 1×31 \times 3

The product PQPQ will be a 3×33 \times 3 matrix given by:

PQ=[−1\2\1]×[2−41]=[−24−1\4−82\2−41]PQ = \begin{bmatrix}-1 \\\2 \\\1\end{bmatrix}\times\begin{bmatrix}2 & -4 & 1\end{bmatrix}=\begin{bmatrix}-2 & 4 & -1 \\\4 & -8 & 2 \\\2 & -4 & 1\end{bmatrix}

Next, we find the transpose (PQ)′(PQ)':

(PQ)′=[−24−1\4−82\2−41]′=[−242\4−8−4−121](PQ)' = \begin{bmatrix}-2 & 4 & -1 \\\4 & -8 & 2 \\\2 & -4 & 1\end{bmatrix}'=\begin{bmatrix}-2 & 4 & 2 \\\4 & -8 & -4 \\\\-1 & 2 & 1\end{bmatrix}

Thus, the correct answer is:

[−242\4−8−4−121]\begin{bmatrix}-2 & 4 & 2 \\\4 & -8 & -4 \\\\-1 & 2 & 1\end{bmatrix}