If then for n ≥ 4, An =

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

If $A = [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}]$ then for n ≥ 4, Aⁿ =

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$A = [\begin{array}{lll} 1 0 0 \\ 1 0 1 \\ 0 1 0 \end{array}]$

$A^{2} = [\begin{array}{lll} 1 0 0 \\ 1 0 1 \\ 0 1 0 \end{array}] [\begin{array}{lll} 1 0 0 \\ 1 0 1 \\ 0 1 0 \end{array}] = [\begin{array}{lll} 1 0 0 \\ 1 1 0 \\ 1 0 1 \end{array}]$

$A^{4} = [\begin{array}{lll} 1 0 0 \\ 1 1 0 \\ 1 0 1 \end{array}] [\begin{array}{lll} 1 0 0 \\ 1 1 0 \\ 1 0 1 \end{array}] = [\begin{array}{lll} 1 0 0 \\ 2 1 0 \\ 2 0 1 \end{array}]$

$A^{5} = [\begin{array}{lll} 1 0 0 \\ 2 1 0 \\ 2 0 1 \end{array}] [\begin{array}{lll} 1 0 0 \\ 1 0 1 \\ 0 1 0 \end{array}] = [\begin{array}{lll} 1 0 0 \\ 3 0 1 \\ 2 1 0 \end{array}]$

$A^{6} = [\begin{array}{lll} 1 0 0 \\ 3 0 1 \\ 2 1 0 \end{array}] [\begin{array}{lll} 1 0 0 \\ 1 0 1 \\ 0 1 0 \end{array}] = [\begin{array}{lll} 1 0 0 \\ 3 1 0 \\ 3 0 1 \end{array}]$

Continuing in this way, we get

$A^{n} = [\begin{array}{lll} 0 0 0 \\ n / 2 1 0 \\ n / 2 0 1 \end{array}],$ when n is even

$A^{n} = [\begin{array}{ccc} 1 & 0 & 0 \\ n + 1 / 2 & 0 & 1 \\ n - 1 / 2 & 1 & 0 \end{array}]$ , when n is odd

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