If A - - 0 6 7- -6 0 8- 7 -8 0 - B - - 0 1 1- 1 0 2- 1 2 0 - C - - 2- -2- 3 - calculate AC- BC and A - BC-Also- verify that A - BC - AC - BC-

If AC = [ 0 #160; amp; 6 amp; 7; 6 #160; amp; 0 amp; 8; 7 #160; amp; 8 amp; 0 ][ #160; 2; 2 #160;; 3 #160; ] ...

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Byju's Answer

Standard XII

Mathematics

Multiplication of Matrices

If A = [ 0 ...

Question

If $A = ⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 7 - 6 & 0 & 8 7 & - 8 & 0 \end{matrix} ⎤ ⎥ ⎦, B = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 1 1 & 0 & 2 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦, C = ⎡ ⎢ ⎣ \begin{matrix} 2 - 2 3 \end{matrix} ⎤ ⎥ ⎦$ , calculate $A C, B C$ and $(A + B) C$ .
Also, verify that $(A + B) C = A C + B C$ .

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Solution

If $A C = ⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 7 - 6 & 0 & 8 7 & - 8 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 2 - 2 3 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 0 - 12 + 21 - 12 + 0 + 24 14 + 16 + 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 91230 \end{matrix} ⎤ ⎥ ⎦$
$B C = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 1 1 & 0 & 2 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 2 - 2 3 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 0 - 2 + 3 2 - 0 + 6 2 - 4 + 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 18 - 2 \end{matrix} ⎤ ⎥ ⎦$
$A + B = ⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 7 - 6 & 0 & 8 7 & - 8 & 0 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 1 1 & 0 & 2 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 0 & 7 & 8 - 5 & 0 & 10 8 & - 6 & 0 \end{matrix} ⎤ ⎥ ⎦$
$(A + B) C = ⎡ ⎢ ⎣ \begin{matrix} 0 & 7 & 8 - 5 & 0 & 10 8 & - 6 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 2 - 2 3 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 0 - 14 + 24 10 + 0 + 30 16 + 12 + 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 102028 \end{matrix} ⎤ ⎥ ⎦ . . . . (1)$
$A C + B C = ⎡ ⎢ ⎣ \begin{matrix} 91230 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 18 - 2 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 102028 \end{matrix} ⎤ ⎥ ⎦ . . . (2)$
From equations $(1)$ and $(2)$
$(A + B) C = A C + B C$ (Proved).

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If A=⎡⎢⎣067−6087−80⎤⎥⎦,B=⎡⎢⎣011102120⎤⎥⎦,C=⎡⎢⎣2−23⎤⎥⎦, calculate AC,BC and (A+B)C.Also, verify that (A+B)C=AC+BC.