Whether the given matrix is symmetric, skew-symmetric or orthogonal and find the spectrum of matrix. The given matrix A is orthogonal _ if and only if a 2 + b 2 = 1 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 = [ a b − b a ] . Check whether the matrix A is symmetric or not: A T = [ a b − b a ] T = [ a − b b a ] A T ≠ A Thus, the given matrix A is not symmetric. Check whether the matrix A is skew-symmetric or not: A T = [ a b − b a ] T = [ a − b b a ] A T ≠ − A Thus, the given matrix A is not skew-symmetric. Check whether the matrix A is orthogonal or not: A T A = [ a − b b a ] [ a b − b a ] = [ a 2 + b 2 a b − b a a b − b a a 2 + b 2 ] = [ a 2 + b 2 0 0 a 2 + b 2 ] The matrix A is orthogonal _ if and only if a 2 + b 2 = 1 . Obtain the spectrum of matrix as follows. | A − λ I | = 0 | a − λ b − b a − λ | = 0 ⇒ ( a − λ ) 2 + b 2 = 0 ⇒ λ 2 − 2 a λ + a 2 + b 2 = 0 λ = − ( − 2 a ) ± ( − 2 a ) 2 − 4 ( 1 ) ( a 2 + b 2 ) 2 ( 1 ) Simplify further as follows. λ = 2 a ± 4 a 2 − 4 a 2 − 4 b 2 2 = 2 a ± − 4 b 2 2 = a ± i b | λ | = a 2 + b 2 = 1 ( if orthogonal ) Thus, the spectrum of matrix is { 1 } _ .

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