Question

5.3x-y-2z = 22y-z=-13x-5y = 3

Answer 1

The given system of equations is,

3x−y−2z=2 2y−z=−1 3x−5y=3

Write the system of equations in the form of AX=B.

[ 3 −1 −2 0 2 −1 3 −5 0 ][ x y z ]=[ 2 −1 3 ]

Now, the determinant of A is,

| A |=3( 2×0−1×5 )+1( 0×0+1×3 )−2( −5×0−3×2 ) =−15+3+12 =0

Since | A |=0, so A is a singular matrix.

The co-factors of the each elements of the matrix are,

A 11 = ( −1 ) 1+1 [ 2×0−( −5 )( −1 ) ] =−5

A 12 = ( −1 ) 1+2 [ 0×0−3( −1 ) ] =−3

A 13 = ( −1 ) 1+3 [ −5×0−3×2 ] =−6

A 21 = ( −1 ) 2+1 [ −1×0−( −2 )×( −5 ) ] =10

A 22 = ( −1 ) 2+2 [ 3×0−( −2 )×3 ] =6

A 23 = ( −1 ) 2+3 [ −5×3−3×( −1 ) ] =−( −12 ) =12

A 31 = ( −1 ) 3+1 [ ( −1 )( −1 )−2( −2 ) ] =5

A 32 = ( −1 ) 3+2 [ −1×3−( −2 )×0 ] =−( −3 ) =3

A 33 = ( −1 ) 3+3 [ 3×2−0×( −1 ) ] =6

Thus, the value of ( adjA )( B ) is,

( adjA )( B )=[ −5 10 5 −3 6 3 −6 12 6 ][ 2 −1 3 ] =[ −10−10+15 −6−6+9 −12−12+18 ] =[ −5 −3 −6 ] ≠0

The solution of the given system of equation does not exist. Hence, the system of equations is inconsistent.