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Let $A=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$ and $C=\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]$ Find each of the following:
(i) $\mathbf{A}+\mathbf{B}$
(ii) $\mathbf{A}-\mathbf{B}$
(iii) $3 \mathbf{A}-\mathrm{C}$
(iv) $\mathrm{AB}$
(v) $\mathbf{B A}$
(i) $\mathbf{A}+\mathbf{B}$
(ii) $\mathbf{A}-\mathbf{B}$
(iii) $3 \mathbf{A}-\mathrm{C}$
(iv) $\mathrm{AB}$
(v) $\mathbf{B A}$
Solution:
2980 Upvotes
Verified Answer
(i) $\mathrm{A}+\mathrm{B}=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]+\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]=\left[\begin{array}{ll}3 & 7 \\ 1 & 7\end{array}\right]$
(ii) $\mathrm{A}-\mathrm{B}=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]-\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]=\left[\begin{array}{cc}1 & 1 \\ 5 & -3\end{array}\right]$
(iii) $3 \mathrm{~A}-\mathrm{C}=3\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]-\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]=\left[\begin{array}{ll}8 & 7 \\ 6 & 2\end{array}\right]$
(iv) $\mathrm{AB}=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$
$$
=\left[\begin{array}{ll}
2-8 & 6+20 \\
3-4 & 9+10
\end{array}\right]=\left[\begin{array}{ll}
-6 & 26 \\
-1 & 19
\end{array}\right]
$$
(v) $\mathrm{BA}=\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]=\left[\begin{array}{cc}11 & 10 \\ 11 & 2\end{array}\right]$
(ii) $\mathrm{A}-\mathrm{B}=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]-\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]=\left[\begin{array}{cc}1 & 1 \\ 5 & -3\end{array}\right]$
(iii) $3 \mathrm{~A}-\mathrm{C}=3\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]-\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]=\left[\begin{array}{ll}8 & 7 \\ 6 & 2\end{array}\right]$
(iv) $\mathrm{AB}=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$
$$
=\left[\begin{array}{ll}
2-8 & 6+20 \\
3-4 & 9+10
\end{array}\right]=\left[\begin{array}{ll}
-6 & 26 \\
-1 & 19
\end{array}\right]
$$
(v) $\mathrm{BA}=\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]=\left[\begin{array}{cc}11 & 10 \\ 11 & 2\end{array}\right]$
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