Whether the matrix A = [ 0 0 1 0 1 0 − 1 0 0 ] 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 0 1 0 1 0 − 1 0 0 ] . Check whether the matrix A is symmetric or not as follows. A T = [ 0 0 1 0 1 0 − 1 0 0 ] T = [ 0 0 − 1 0 1 0 1 0 0 ] A T ≠ A Thus, the given matrix A is not symmetric. Check whether the matrix A is skew-symmetric or not as follows. A T = [ 0 0 1 0 1 0 − 1 0 0 ] T = [ 0 0 − 1 0 1 0 1 0 0 ] A T ≠ − A Thus, the given matrix A is not skew-symmetric. Check whether the matrix A is orthogonal or not as follows. A T A = [ 0 0 − 1 0 1 0 1 0 0 ] [ 0 0 1 0 1 0 − 1 0 0 ] = [ 1 0 0 0 1 0 0 0 1 ] = I Thus, the given matrix A is orthogonal _ . Obtain the spectrum of matrix as follows. | A − λ I | = 0 | − λ 0 1 0 1 − λ 0 − 1 0 − λ | = 0 ( − λ ) [ ( − λ ) ( 1 − λ ) ] + 1 [ 1 ⋅ ( 1 − λ ) ] = 0 ( λ 2 + 1 ) ( 1 − λ ) = 0 λ = 1 or λ = ± i On further simplifications, the following is obtained. | λ | = 1 or | λ | = 1 Thus, the spectrum of matrix is { 1 } _ .

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