If , satisfies the matrix equation A2 = kA, write the value of k.

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

If $A = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$ , satisfies the matrix equation A² = kA, write the value of k.

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$A = [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}]$

\Rightarrow A^{2} = [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}] [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}]

\Rightarrow A^{2} = [\begin{array}{cc} 1 + 1 & - 1 - 1 \\ - 1 - 1 & 1 + 1 \end{array}]

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

Now,

A² = kA

\Rightarrow [\begin{array}{cc} 2 & - 2 \\ - 2 & 2 \end{array}] = k [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}]

\Rightarrow [\begin{array}{cc} 2 & - 2 \\ - 2 & 2 \end{array}] = [\begin{array}{cc} k & - k \\ - k & k \end{array}]

⇒ k = 2

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