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Question

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( 3×25×3 ) =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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