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# 00-131358

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Solution

## The given matrix is, A=[ 1 0 0 3 3 0 5 2 −1 ] The inverse of a matrix exists only if it satisfies the condition of | A |≠0. So, the determinant of A is, | A |=1[ 3×( −1 )−2×0 ]−0( 3×( −1 )−5×0 )+0( 3×2−5×3 ) =−3 Since, | A |≠0, so inverse of the matrix A exists. The formula to calculate the inverse of the matrix is, A −1 = adjA | A | (1) The cofactors of each element of the matrix is, A 11 = ( −1 ) 1+1 [ 3×( −1 )−( 2×0 ) ] =−3 A 12 = ( −1 ) 1+2 [ 3×( −1 )−( 5×0 ) ] =−( −3 ) =3 A 13 = ( −1 ) 1+3 [ ( 3×2 )−( 5×3 ) ] =−9 A 21 = ( −1 ) 2+1 [ 0×( −1 )−( 0×2 ) ] =0 A 22 = ( −1 ) 2+2 [ 1×( −1 )−( 0×5 ) ] =−1 A 23 = ( −1 ) 2+3 [ ( 1×2 )−( 5×0 ) ] =−2 A 31 = ( −1 ) 3+1 [ ( 0×0 )−( 3×0 ) ] =0 A 32 = ( −1 ) 3+2 [ ( 1×0 )−( 3×0 ) ] =0 A 33 = ( −1 ) 3+3 [ ( 1×3 )−( 3×0 ) ] =3 The adjoint of A will be, adjA=[ A 11 A 21 A 31 A 12 A 22 A 32 A 13 A 23 A 33 ] =[ −3 0 0 3 −1 0 −9 −2 3 ] Substitute [ −3 0 0 3 −1 0 −9 −2 3 ] for adjA and −3 for | A | in equation (1), A −1 =− 1 3 [ −3 0 0 3 −1 0 −9 −2 3 ]

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