Given that and .Verify that AB = AC though B ≠ C, A ≠ 0.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Multiplication of Matrices

Question

Given that $A = [\begin{array}{ccc} 1 & 1 & 1 \\ 3 & 3 & 3 \end{array}], B = [\begin{array}{cc} 3 & 1 \\ 5 & 2 \\ - 2 & 4 \end{array}]$ and $C = [\begin{array}{cc} 4 & 2 \\ - 3 & 5 \\ 5 & 0 \end{array}]$ .
Verify that AB = AC though B ≠ C, A ≠ 0.

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$A = [\begin{array}{ccc} 1 & 1 & 1 \\ 3 & 3 & 3 \end{array}], B = [\begin{array}{cc} 3 & 1 \\ 5 & 2 \\ - 2 & 4 \end{array}], C = [\begin{array}{cc} 4 & 2 \\ - 3 & 5 \\ 5 & 0 \end{array}]$

AB = [\begin{array}{ccc} 1 & 1 & 1 \\ 3 & 3 & 3 \end{array}] \times [\begin{array}{cc} 3 & 1 \\ 5 & 2 \\ - 2 & 4 \end{array}]

= [\begin{array}{cc} 3 + 5 - 2 & 1 + 2 + 4 \\ 9 + 15 - 6 & 3 + 6 + 12 \end{array}] = [\begin{array}{cc} 6 & 7 \\ 18 & 21 \end{array}]

AC = [\begin{array}{ccc} 1 & 1 & 1 \\ 3 & 3 & 3 \end{array}] \times [\begin{array}{cc} 4 & 2 \\ - 3 & 5 \\ 5 & 0 \end{array}]

= [\begin{array}{cc} 4 - 3 + 5 & 2 + 5 - 0 \\ 12 - 9 + 15 & 6 + 15 + 0 \end{array}] = [\begin{array}{cc} 6 & 7 \\ 18 & 21 \end{array}]

∴ AB = BC