If , show that A'A = I.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Multiplication of Matrices

Question

If $A = [\begin{array}{cc} \cos α & \sin α \\ - \sin α & \cos α \end{array}]$ , show that A'A = I.

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$A = [\begin{array}{cc} \cos α & \sin α \\ - \sin α & \cos α \end{array}]$

on transposing it,

$A^{'} = [\begin{array}{cc} \cos α & - \sin α \\ \sin α & \cos α \end{array}]$

Now ${AA}^{'} = [\begin{array}{cc} \cos α & \sin α \\ - \sin α & \cos α \end{array}] [\begin{array}{cc} \cos α & - \sin α \\ \sin α & \cos α \end{array}]$

${AA}^{'} = [\begin{array}{lc} \cos^{2} α + \sin^{2} α & - \cos αsin α + \sin αcos α \\ - \sin αcos α + \cos αsin α & \sin^{2} α + \cos^{2} α \end{array}]$

${AA}^{'} = [\begin{array}{ll} 1 0 \\ 0 1 \end{array}]$

AA' = I