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-

Gven: nbsp;A = nbsp;[ 0 amp;6 amp;7; nbsp; 6 amp;0 amp;8; 7 amp; 8 amp;0 ], B = nbsp;[ 0 amp;1 amp;1; nbsp; 1 ...

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Byju's Answer

Standard XII

Mathematics

Consistency of Linear System of Equations

If A = [ 0 6 ...

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

Gven: $A = ⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 7 - 6 & 0 & 8 7 & - 8 & 0 \end{matrix} ⎤ ⎥ ⎦, B = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 1 1 & 0 & 2 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦, C = ⎡ ⎢ ⎣ \begin{matrix} 2 - 2 3 \end{matrix} ⎤ ⎥ ⎦$
Calculating $A C$
$A C = {⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 7 - 6 & 0 & 8 7 & - 8 & 0 \end{matrix} ⎤ ⎥ ⎦}_{3 \times 3} {⎡ ⎢ ⎣ \begin{matrix} 2 - 2 3 \end{matrix} ⎤ ⎥ ⎦}_{3 \times 1}$
$A C = ⎡ ⎢ ⎣ \begin{matrix} 0 (2) + 6 (- 2) + 7 (3) - 6 (2) + 0 (- 2) + 8 (3) 7 (2) - 8 (- 2) + 0 (3) \end{matrix} ⎤ ⎥ ⎦$
$A C = ⎡ ⎢ ⎣ \begin{matrix} 0 - 12 + 21 - 12 + 0 + 24 14 + 16 + 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 91230 \end{matrix} ⎤ ⎥ ⎦$
Calculating $B C$
$B C = {⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 1 1 & 0 & 2 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦}_{3 \times 3} {⎡ ⎢ ⎣ \begin{matrix} 2 - 2 3 \end{matrix} ⎤ ⎥ ⎦}_{3 \times 1}$
$B C = {⎡ ⎢ ⎣ \begin{matrix} 0 (2) + 1 (- 2) + 1 (3) 1 (2) + 0 (- 2) + 2 (3) 1 (2) + 2 (- 2) + 0 (3) \end{matrix} ⎤ ⎥ ⎦}_{3 \times 1}$
$B C = ⎡ ⎢ ⎣ \begin{matrix} 0 - 2 + 3 2 + 0 + 6 2 - 4 + 0 \end{matrix} ⎤ ⎥ ⎦$
$B C = ⎡ ⎢ ⎣ \begin{matrix} 18 - 2 \end{matrix} ⎤ ⎥ ⎦$
Calculating $(A + B) C$
$(A + B) C = ⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 0 + 0 & 6 + 1 & 7 + 1 - 6 + 1 & 0 + 0 & 8 + 2 7 + 1 & - 8 + 2 & 0 + 0 \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠ C$
$(A + B) C = {⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 0 & 7 & 8 - 5 & 0 & 10 8 & - 6 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠}_{3 \times 3} {⎡ ⎢ ⎣ \begin{matrix} 2 - 2 3 \end{matrix} ⎤ ⎥ ⎦}_{3 \times 1}$
$(A + B) C = {⎡ ⎢ ⎣ \begin{matrix} 0 (2) + 7 (- 2) + 8 (3) - 5 (2) + 0 (- 2) + 10 (3) 8 (2) + (- 6) (- 2) + 0 (3) \end{matrix} ⎤ ⎥ ⎦}_{3 \times 1}$
$(A + B) C = ⎡ ⎢ ⎣ \begin{matrix} 0 - 14 + 24 - 10 + 0 + 30 16 + 12 + 0 \end{matrix} ⎤ ⎥ ⎦$
$(A + B) C = ⎡ ⎢ ⎣ \begin{matrix} 102028 \end{matrix} ⎤ ⎥ ⎦$
Solve for proving $(A + B) C = A C + B C$
Calculating $A C + B C$
$A C + B C = ⎡ ⎢ ⎣ \begin{matrix} 91230 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 18 - 2 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 9 + 1 12 + 8 30 - 2 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 102028 \end{matrix} ⎤ ⎥ ⎦$
So, $(A + B) C = A C + B C$
$∴ L . H . S . = R . H . S .$
Hence proved.

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Similar questions

Q. 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

Q. If

A = [\begin{matrix} 1 & 2 2 & 1 \end{matrix}] . B = [\begin{matrix} 2 & 0 1 & 3 \end{matrix}]

and

C = [\begin{matrix} 1 & 1 2 & 3 \end{matrix}]

calculate

A C, B C

and

(A + B) C

. Also verify that

(A + B) C = A C + B C .

Using elementary row operations (transformations), find the inverse of $⎛ ⎜ ⎝ \begin{matrix} 0 & 1 & 2 1 & 2 & 3 3 & 1 & 0 \end{matrix} ⎞ ⎟ ⎠$ OR 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= $⎡ ⎢ ⎣ \begin{matrix} 2 - 2 3 \end{matrix} ⎤ ⎥ ⎦$ , then calculate AC, BC and (A+B) C.