Whether the given matrix is symmetric, skew-symmetric or orthogonal and find the spectrum of matrix. The given matrix A is orthogonal _ and the spectrum of matrix is { 1 } _ . Definition used: 1. A real square matrix A = [ a j k ] is called symmetric if transposition leaves it unchanged A T = A . 2. A real square matrix A = [ a j k ] is called skew-symmetric if transposition gives the negative of A , A T = − A . 3. A real square matrix A = [ a j k ] is called orthogonal if transposition gives the inverse of A , A T = A − 1 . Calculation: The given matrix is A = [ 0.8 0.6 − 0.6 0.8 ] . Check whether the matrix A is symmetric or not: A T = [ 0.8 0.6 − 0.6 0.8 ] T = [ 0.8 − 0.6 0.6 0.8 ] A T ≠ A Thus, the given matrix A is not symmetric. Check whether the matrix A is skew-symmetric or not: A T = [ 0.8 0.6 − 0.6 0.8 ] T = [ 0.8 − 0.6 0.6 0.8 ] A T ≠ − A Thus, the given matrix A is not skew-symmetric. Check whether the matrix A is orthogonal or not: A T A = [ 0.8 − 0.6 0.6 0.8 ] [ 0.8 0.6 − 0.6 0.8 ] = [ ( 0.8 ) 2 + ( 0.6 ) 2 ( 0.8 ) ( 0.6 ) − ( 0.6 ) ( 0.8 ) ( 0.6 ) ( 0.8 ) − ( 0.8 ) ( 0.6 ) ( 0.8 ) 2 + ( 0.6 ) 2 ] = [ 1 0 0 1 ] Thus, the given matrix A is orthogonal _ . Obtain the spectrum of matrix as follows. | A − λ I | = 0 | 0.8 − λ 0.6 − 0.6 0.8 − λ | = 0 ⇒ ( 0.8 − λ ) 2 + ( 0.6 ) 2 = 0 ⇒ λ 2 − 1.6 λ + 1 = 0 λ = − ( − 1.6 ) ± ( − 1.6 ) 2 − 4 ( 1 ) ( 1 ) 2 ( 1 ) Simplify further as follows. λ = 1.6 ± 2.56 − 4 2 = 1.6 ± − 1.44 2 = 0.8 ± i 0.6 | λ | = ( 0.8 ) 2 + ( 0.6 ) 2 = 1 Thus, the spectrum of matrix is { 1 } _ .

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