Let A = cos θ 0 sin θ 0 1 0 − sin θ 0 cos θ . If for some θ ∈ ( 0 , π ) , A 2 = A T , then the sum of the diagonal elements of the matrix ( A + I ) 3 + ( A − I ) 3 − 6 A is equal to _____ .

Question

Let A = ​ cos θ 0 sin θ ​ 0 1 0 ​ − sin θ 0 cos θ ​ ​ . If for some θ ∈ ( 0 , π ) , A 2 = A T , then the sum of the diagonal elements of the matrix ( A + I ) 3 + ( A − I ) 3 − 6 A is equal to _____ .

TestHub · Accepted Answer

∵ A is orthogonal matrix ​ ∴ A T = A − 1 ⇒ A 2 = A − 1 ( ∵ A 2 = A T ) ⇒ A 3 = I let B = ( A + I ) 3 + ( A − I ) 3 − 6 A = 2 ( A 3 + 3 A ) − 6 A = 2 A 3 B = 2 I = ​ 2 0 0 ​ 0 2 0 ​ 0 0 2 ​ ​ ​ Now sum of diagonal elements = 2 + 2 + 2 = 6

Mathematics - Matrices Question with Solution | TestHub

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