Let

Question:

Let $A=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{rr}1 & 3 \\ -2 & 5\end{array}\right]$ and $C=\left[\begin{array}{rr}-2 & 5 \\ 3 & 4\end{array}\right]$. Find each of the following:

(i) $2 A-3 B$

(ii) $B-4 C$

(iii) $3 A-C$

(iv) $3 A-2 B+3 C$

Solution:

(i) $2 A-3 B$

$\Rightarrow 2 A-3 B=2\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]-3\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$

$\Rightarrow 2 A-3 B=\left[\begin{array}{ll}4 & 8 \\ 6 & 4\end{array}\right]-\left[\begin{array}{cc}3 & 9 \\ -6 & 15\end{array}\right]$

$\Rightarrow 2 A-3 B=\left[\begin{array}{cc}4-3 & 8-9 \\ 6+6 & 4-15\end{array}\right]$

$\Rightarrow 2 A-3 B=\left[\begin{array}{cc}1 & -1 \\ 12 & -11\end{array}\right]$

(ii) $B-4 C$

$\Rightarrow B-4 C=\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]-4\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]$

$\Rightarrow B-4 C=\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]-\left[\begin{array}{cc}-8 & 20 \\ 12 & 16\end{array}\right]$

$\Rightarrow B-4 C=\left[\begin{array}{cc}1+8 & 3-20 \\ -2-12 & 5-16\end{array}\right]$

$\Rightarrow B-4 C=\left[\begin{array}{cc}9 & -17 \\ -14 & -11\end{array}\right]$

(iii) $3 A-C$

$\Rightarrow 3 A-C=3\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]-\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]$

$\Rightarrow 3 A-C=\left[\begin{array}{cc}6 & 12 \\ 9 & 6\end{array}\right]-\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]$

$\Rightarrow 3 A-C=\left[\begin{array}{ll}6+2 & 12-5 \\ 9-3 & 6-4\end{array}\right]$

$\Rightarrow 3 A-C=\left[\begin{array}{ll}8 & 7 \\ 6 & 2\end{array}\right]$

(iv) $3 A-2 B+3 C$

$\Rightarrow 3 A-2 B+3 C=3\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]-2\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]+3\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]$

$\Rightarrow 3 A-2 B+3 C=\left[\begin{array}{cc}6 & 12 \\ 9 & 6\end{array}\right]-\left[\begin{array}{cc}2 & 6 \\ -4 & 10\end{array}\right]+\left[\begin{array}{cc}-6 & 15 \\ 9 & 12\end{array}\right]$

$\Rightarrow 3 A-2 B+3 C=\left[\begin{array}{cc}6-2-6 & 12-6+15 \\ 9+4+9 & 6-10+12\end{array}\right]$

$\Rightarrow 3 A-2 B+3 C=\left[\begin{array}{cc}-2 & 21 \\ 22 & 8\end{array}\right]$

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