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Question

Evaluate the determinants (i) (iii) (ii) (iv)

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Solution

(i)

The given matrix is,

[ 3 1 2 0 0 1 3 5 0 ]

Determinant of the given matrix is,

| 3 1 2 0 0 1 3 5 0 | =3| 0 1 5 0 |( 1 )| 0 1 3 0 |+( 2 )| 0 0 3 5 | =3( 0×0( 1 )×( 5 ) )+1( 0+3 )2( 0×( 5 )0×3 ) =12

Thus, the value of the determinant is 12.

(ii)

The given matrix is,

[ 3 4 5 1 1 2 2 3 1 ]

Determinant of the given matrix is,

| 3 4 5 1 1 2 2 3 1 | =3| 1 2 3 1 |( 4 )| 1 2 2 1 |+5| 1 1 2 3 | =3( 1×1( 2 )×( 3 ) )+4( 1( 2 )×2 )5( 1×( 3 )1×2 ) =46

Thus, the value of the determinant is 46.

(iii)

The given matrix is,

[ 0 1 2 1 0 3 2 3 0 ]

Determinant of the given matrix is,

| 0 1 2 1 0 3 2 3 0 | =0| 0 3 3 0 |( 1 )| 1 3 2 0 |+( 2 )| 1 0 2 3 | =0( 0×0( 3 )×( 3 ) )1( 06 )2( 3×( 1 )0×( 2 ) ) =0

Thus, the value of determinant is 0.

(iv)

The given matrix is,

[ 2 1 2 0 2 1 3 5 0 ]

Determinant of the given matrix is,

| 2 1 2 0 2 1 3 5 0 | =2| 2 1 5 0 |( 1 )| 0 1 3 0 |+( 2 )| 0 2 3 5 | =2( 2×0( 1 )×( 5 ) )+1( 0+3 )2( 0×( 5 )3×( 2 ) ) =5

Thus, the value of the determinant is 5.


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