- If A-1 0-2 3-1 0-2 1 1- B- 05-4-21 3-10 2 and C-1 52-1 10 0-11- verify that AB-C-A B-A C - -

Given: #160;AB C=AB AC #8658;10 23 10 21105 4 213 102 152 1100 11=10 23 10 21105 4 213 102 10 23 10 211152 1100 11 #8658;10 23 10 2110 15 5 4 2 2+11 13 0 ...

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Standard IV

Mathematics

Multiplication Trick for a Table

[ ...

Question

If $A = [\begin{matrix} 1 & 0 & - 2 \\ 3 & - 1 & 0 \\ - 2 & 1 & 1 \end{matrix}], B = [\begin{matrix} 0 & 5 & - 4 \\ - 2 & 1 & 3 \\ - 1 & 0 & 2 \end{matrix}] and C = [\begin{matrix} 1 & 5 & 2 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{matrix}]$ , verify that A (B − C) = AB − AC.

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Solution

$Given : A (B - C) = A B - A C \Rightarrow [\begin{matrix} 1 & 0 & - 2 \\ 3 & - 1 & 0 \\ - 2 & 1 & 1 \end{matrix}] ([\begin{matrix} 0 & 5 & - 4 \\ - 2 & 1 & 3 \\ - 1 & 0 & 2 \end{matrix}] - [\begin{matrix} 1 & 5 & 2 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{matrix}]) = [\begin{matrix} 1 & 0 & - 2 \\ 3 & - 1 & 0 \\ - 2 & 1 & 1 \end{matrix}] [\begin{matrix} 0 & 5 & - 4 \\ - 2 & 1 & 3 \\ - 1 & 0 & 2 \end{matrix}] - [\begin{matrix} 1 & 0 & - 2 \\ 3 & - 1 & 0 \\ - 2 & 1 & 1 \end{matrix}] [\begin{matrix} 1 & 5 & 2 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{matrix}] \Rightarrow [\begin{matrix} 1 & 0 & - 2 \\ 3 & - 1 & 0 \\ - 2 & 1 & 1 \end{matrix}] [\begin{matrix} 0 - 1 & 5 - 5 & - 4 - 2 \\ - 2 + 1 & 1 - 1 & 3 - 0 \\ - 1 - 0 & 0 + 1 & 2 - 1 \end{matrix}] = [\begin{matrix} 0 - 0 + 2 & 5 + 0 - 0 & - 4 + 0 - 4 \\ 0 + 2 - 0 & 15 - 1 + 0 & - 12 - 3 + 0 \\ 0 - 2 - 1 & - 10 + 1 + 0 & 8 + 3 + 2 \end{matrix}] - [\begin{matrix} 1 - 0 - 0 & 5 + 0 + 2 & 2 + 0 - 2 \\ 3 + 1 + 0 & 15 - 1 - 0 & 6 - 0 + 0 \\ - 2 - 1 + 0 & - 10 + 1 - 1 & - 4 + 0 + 1 \end{matrix}] \Rightarrow [\begin{matrix} 1 & 0 & - 2 \\ 3 & - 1 & 0 \\ - 2 & 1 & 1 \end{matrix}] [\begin{matrix} - 1 & 0 & - 6 \\ - 1 & 0 & 3 \\ - 1 & 1 & 1 \end{matrix}] = [\begin{matrix} 2 & 5 & - 8 \\ 2 & 14 & - 15 \\ - 3 & - 9 & 13 \end{matrix}] - [\begin{matrix} 1 & 7 & 0 \\ 4 & 14 & 6 \\ - 3 & - 10 & - 3 \end{matrix}] \Rightarrow [\begin{matrix} - 1 - 0 + 2 & 0 + 0 - 2 & - 6 + 0 - 2 \\ - 3 + 1 - 0 & 0 - 0 + 0 & - 18 - 3 + 0 \\ 2 - 1 - 1 & 0 + 0 + 1 & 12 + 3 + 1 \end{matrix}] = [\begin{matrix} 2 - 1 & 5 - 7 & - 8 - 0 \\ 2 - 4 & 14 - 14 & - 15 - 6 \\ - 3 + 3 & - 9 + 10 & 13 + 3 \end{matrix}] \Rightarrow [\begin{matrix} 1 & - 2 & - 8 \\ - 2 & 0 & - 21 \\ 0 & 1 & 16 \end{matrix}] = [\begin{matrix} 1 & - 2 & - 8 \\ - 2 & 0 & - 21 \\ 0 & 1 & 16 \end{matrix}] ∴ LHS = RHS Hence proved .$

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If A= 1 0-2 3-1 0-2 1 1, B= 05-4-21 3-10 2 and C= 1 52-1 10 0-11, verify that A (B − C) = AB − AC.

If $A = [\begin{matrix} 1 & 0 & - 2 \\ 3 & - 1 & 0 \\ - 2 & 1 & 1 \end{matrix}], B = [\begin{matrix} 0 & 5 & - 4 \\ - 2 & 1 & 3 \\ - 1 & 0 & 2 \end{matrix}] and C = [\begin{matrix} 1 & 5 & 2 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{matrix}]$ , verify that A (B − C) = AB − AC.