If , then find A4.

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

If $A = [\begin{array}{cc} - 3 & 0 \\ 0 & - 3 \end{array}]$ , then find A⁴.

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$A = [\begin{array}{cc} - 3 & 0 \\ 0 & - 3 \end{array}]$

A^{2} = [\begin{array}{cc} - 3 & 0 \\ 0 & - 3 \end{array}] [\begin{array}{cc} - 3 & 0 \\ 0 & - 3 \end{array}]

= [\begin{array}{cc} 9 + 0 & 0 + 0 \\ 0 + 0 & 0 + 9 \end{array}]

\Rightarrow A^{2} = [\begin{array}{cc} 9 & 0 \\ 0 & 9 \end{array}]

⇒ A⁴ = A².A²

= [\begin{array}{cc} 9 & 0 \\ 0 & 9 \end{array}] [\begin{array}{cc} 9 & 0 \\ 0 & 9 \end{array}]

= [\begin{array}{cc} 81 + 0 & 0 + 0 \\ 0 + 0 & 0 + 81 \end{array}]

∴ A^{4} = [\begin{array}{cc} 81 & 0 \\ 0 & 81 \end{array}]

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