If A- 2 3- 0 5 - B- -7 -3- 2 4 - C- -1 0- -4 7 - verify that A-2B-3C-A-2B-3C-

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Byju's Answer

Standard XII

Chemistry

Liquids in Liquids

If A=[[ 2 3; ...

Question

If $A = [\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}], B = [\begin{matrix} - 7 & - 3 \\ 2 & 4 \end{matrix}], C = [\begin{matrix} - 1 & 0 \\ - 4 & 7 \end{matrix}]$ , verify that $(A - 2 B) + 3 C = A - (2 B - 3 C)$ .

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Solution

Step1: Calculation of the matrix $(A - 2 B) + 3 C$ .
Multiply each element of $B$ by 2 and each element of $C$ by 3 to obtain $2 B$ and $3 C$ respectively.
$\begin{array}{rcl} (A - 2 B) + 3 C & = & ([\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}] - 2 [\begin{matrix} - 7 & - 3 \\ 2 & 4 \end{matrix}]) + 3 [\begin{matrix} - 1 & 0 \\ - 4 & 7 \end{matrix}] \\ = & ([\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}] - [\begin{matrix} 2 (- 7) & 2 (- 3) \\ 2 \cdot 2 & 2 \cdot 4 \end{matrix}]) + [\begin{matrix} 3 (- 1) & 3 \cdot 0 \\ 3 (- 4) & 3 \cdot 7 \end{matrix}] \\ = & ([\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}] - [\begin{matrix} - 14 & - 6 \\ 4 & 8 \end{matrix}]) + [\begin{matrix} - 3 & 0 \\ - 12 & 21 \end{matrix}] \\ = & [\begin{matrix} 2 - (- 14) & 3 - (- 6) \\ 0 - 4 & 5 - 8 \end{matrix}] + [\begin{matrix} - 3 & 0 \\ - 12 & 21 \end{matrix}] \\ = & [\begin{matrix} 2 + 14 & 3 + 6 \\ - 4 & - 3 \end{matrix}] + [\begin{matrix} - 3 & 0 \\ - 12 & 21 \end{matrix}] \\ = & [\begin{matrix} 16 & 9 \\ - 4 & - 3 \end{matrix}] + [\begin{matrix} - 3 & 0 \\ - 12 & 21 \end{matrix}] \\ = & [\begin{matrix} 16 - 3 & 9 + 0 \\ - 4 - 12 & - 3 + 21 \end{matrix}] \\ = & [\begin{matrix} 13 & 9 \\ - 16 & 18 \end{matrix}] \end{array}$
Step2: Calculation of the matrix $A - (2 B - 3 C)$ .
$\begin{array}{rcl} A - (2 B - 3 C) & = & [\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}] - (2 [\begin{matrix} - 7 & - 3 \\ 2 & 4 \end{matrix}] - 3 [\begin{matrix} - 1 & 0 \\ - 4 & 7 \end{matrix}]) \\ = & [\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}] - ([\begin{matrix} 2 (- 7) & 2 (- 3) \\ 2 \cdot 2 & 2 \cdot 4 \end{matrix}] - [\begin{matrix} 3 (- 1) & 3 \cdot 0 \\ 3 (- 4) & 3 \cdot 7 \end{matrix}]) \\ = & [\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}] - ([\begin{matrix} - 14 & - 6 \\ 4 & 8 \end{matrix}]) - [\begin{matrix} - 3 & 0 \\ - 12 & 21 \end{matrix}] \\ = & [\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}] - [\begin{matrix} - 14 - (- 3) & - 6 - 0 \\ 4 - (- 12) & 8 - 21 \end{matrix}] \\ = & [\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}] - [\begin{matrix} - 14 + 3 & - 6 \\ 4 + 12 & - 13 \end{matrix}] \\ = & [\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}] - [\begin{matrix} - 11 & - 6 \\ 16 & - 13 \end{matrix}] \\ = & [\begin{matrix} 2 - (- 11) & 3 - (- 6) \\ 0 - 16 & 5 - (- 13) \end{matrix}] \\ = & [\begin{matrix} 2 + 11 & 3 + 6 \\ - 16 & 5 + 13 \end{matrix}] \\ = & [\begin{matrix} 13 & 9 \\ - 16 & 18 \end{matrix}] \end{array}$
Conclusion: The relation $(A - 2 B) + 3 C = A - (2 B - 3 C) \begin{array}{l} \end{array}$ is verified.

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If A=2305,B=-7-324,C=-10-47, verify that A-2B+3C=A-2B-3C.

If $A = [\begin{matrix} 2 & 3 \\ 0 & 5 \end{matrix}], B = [\begin{matrix} - 7 & - 3 \\ 2 & 4 \end{matrix}], C = [\begin{matrix} - 1 & 0 \\ - 4 & 7 \end{matrix}]$ , verify that $(A - 2 B) + 3 C = A - (2 B - 3 C)$ .