If A= [ lll 1 & 0 & 1 3 & 1 & 2 ], B= [ lll 2 & 1 & -4 3 & 5 & -2… - Vidyadip

Question

If $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 3 & 1 & 2\end{array}\right], B=\left[\begin{array}{lll}2 & 1 & -4 \\ 3 & 5 & -2\end{array}\right]$ and $C=\left[\begin{array}{ccc}0 & 2 & 3 \\ -1 & -1 & 0\end{array}\right]$, verify that $(A+2 B+3 C)^{\top}=A^{\top}+2 B^{\top}+3 C^{\top}$

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Answer

$\begin{aligned} & A+2 B+3 C \\ & =\left(\begin{array}{lll}1 & 0 & 1 \\ 3 & 1 & 2\end{array}\right)+2\left(\begin{array}{lll}2 & 1 & -4 \\ 3 & 5 & -2\end{array}\right)+3\left(\begin{array}{rrr}0 & 2 & 3 \\ -1 & -1 & 0\end{array}\right) \\ & =\left(\begin{array}{lll}1 & 0 & 1 \\ 3 & 1 & 2\end{array}\right)+\left(\begin{array}{rrr}4 & 2 & -8 \\ 6 & 10 & -4\end{array}\right)+\left(\begin{array}{rrr}0 & 6 & 9 \\ -3 & -3 & 0\end{array}\right) \\ & =\left(\begin{array}{rrr}1+4+0 & 0+2+6 & 1-8+9 \\ 3+6-3 & 1+10-3 & 2-4+0\end{array}\right) \\ & =\left[\begin{array}{rrr}5 & 8 & 2 \\ 6 & 8 & -2\end{array}\right] \\ & \therefore(A+2 B+3 C)^T=\left(\begin{array}{rr}5 & 6 \\ 8 & 8 \\ 2 & -2\end{array}\right) \\ & A^T=\left(\begin{array}{ll}1 & 3 \\ 0 & 1 \\ 1 & 2\end{array}\right), B^T=\left(\begin{array}{rr}2 & 3 \\ 1 & 5 \\ -4 & -2\end{array}\right), C^T=\left(\begin{array}{rr}0 & -1 \\ 2 & -1 \\ 3 & 0\end{array}\right) \\ & \therefore A ^{ T }+2 B ^{ T }+3 C ^{ T } \\ & =\left(\begin{array}{ll}1 & 3 \\ 0 & 1 \\ 1 & 2\end{array}\right)+2\left(\begin{array}{rr}2 & 3 \\ 1 & 5 \\ -4 & -2\end{array}\right)+3\left(\begin{array}{rr}0 & -1 \\ 2 & -1 \\ 3 & 0\end{array}\right) \\ & =\left(\begin{array}{ll}1 & 3 \\ 0 & 1 \\ 1 & 2\end{array}\right)+\left(\begin{array}{rr}4 & 6 \\ 2 & 10 \\ -8 & -4\end{array}\right)+\left(\begin{array}{rr}0 & -3 \\ 6 & -3 \\ 9 & 0\end{array}\right) \\ & =\left(\begin{array}{rr}1+4+0 & 3+6-3 \\ 0+2+6 & 1+10-3 \\ 1-8+9 & 2-4+0\end{array}\right) \\ & =\left[\begin{array}{rr}5 & 6 \\ 8 & 8 \\ 2 & -2\end{array}\right] \\ & \end{aligned}$

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