If and A2 = I. Find the relation given by a2 = I.

Engineering

Mathematics

Algebra of Matrices

Special Type of Square Matrices

Question

If $A = [\begin{array}{cc} α & β \\ γ & - α \end{array}]$ and A² = I. Find the relation given by a² = I.

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$A = [\begin{array}{cc} α & β \\ γ & - α \end{array}] A^{2} = [\begin{array}{cc} α & β \\ γ & - α \end{array}] [\begin{array}{cc} α & β \\ γ & - α \end{array}]$

$A^{2} = [\begin{array}{ll} α^{2} + γβ αγ - αγ \\ βα - αβ βγ + α^{2} \end{array}]$

$= [\begin{array}{cc} α^{2} + γβ & 0 \\ 0 & βγ + α^{2} \end{array}] = [\begin{array}{ll} 1 0 \\ 0 1 \end{array}]$

α² + γβ = 1