If and , show that AB = A and BA = B.

Engineering

Mathematics

Algebra of Matrices

Multiplication of Matrices

Question

If $A = [\begin{array}{ccc} 2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4 \end{array}]$ and $B = [\begin{array}{ccc} 2 & - 2 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 2 & - 3 \end{array}]$ , show that AB = A and BA = B.

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

AB = [\begin{array}{ccc} 4 + 3 - 5 & - 4 - 9 + 10 & - 8 - 12 + 15 \\ - 2 - 4 + 5 & 2 + 12 - 10 & 4 + 16 - 15 \\ 2 + 5 - 4 & - 2 - 9 + 8 & - 4 - 12 + 12 \end{array}]

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

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

BA = [\begin{array}{ccc} 4 + 2 - 4 & - 6 - 8 + 12 & - 10 - 10 + 16 \\ - 2 - 3 + 4 & 3 + 12 - 12 & 5 + 15 - 16 \\ 2 + 2 - 3 & - 3 - 8 + 9 & - 5 - 10 + 12 \end{array}]

BA = [\begin{array}{ccc} 2 & - 2 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 2 & - 3 \end{array}] = B