A square matrix A is called orthogonal if (A^T)A = I_n. A = [[cos(theta), -sin(theta)],[sin(theta), cos(theta)]] Assume that A, B are orthogonal matrices of the same size. Show that AB is also...

A square matrix A is called orthogonal if (A^T)A = I_n.
 
A = [[cos(theta), -sin(theta)],[sin(theta), cos(theta)]]

Assume that A, B are orthogonal matrices of the same size. Show that AB is also orthogonal.

Asked on by bobobooo

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pramodpandey | College Teacher | (Level 3) Valedictorian

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`A=[[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]`

`A^T=[[cos(theta),sin(theta)],[-sin(theta),cos(theta)]]`

`A^T.A=[[cos(theta),sin(theta)],[-sin(theta),cos(theta)]][[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]`

`=[[cos^2(theta)+sin^2(theta),-cos(theta)sin(theta)+sin(theta)cos(theta)],[-sin(theta)cos(theta)+cos(theta)sin(theta),sin^2(theta)+cos^2(theta)]]`

`=[[1,0],[0,1]]`

Thus matrix A is an orthogonal matrix.

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