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Question
Find the inverse of the matrices (if it exists).
`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`
Solution
A = `[(1,0,0),(0,cos alpha, sin alpha),(0,sin alpha, -cos alpha)]`
So, adj `A = [(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]`
`= [(1,0,0),(0,-cos alpha,-sin alpha),(0,-sin alpha,cos alpha)]`
`abs A = 1(- cos^2 alpha = sin alpha) + 0 (0 - 0) + 0 (0 - 0)`
`= -1 ne 0 -> A^-1` exists
`A^-1 = 1/abs A (adj A) = 1/abs A [(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]`
`1/-1 [(1,0,0),(0,-cos alpha,-sin alpha),(0,-sin alpha,cos alpha)]`
`= [(-1,0,0),(0,cos alpha,sin alpha),(0,sin alpha,-cos alpha)]`
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