Question

11. 2x+y+z=13y_5z = 9

Answer 1

The given system of equations is,

2x+y+z=1

x−2y−z= 3 2

3y−5z=9

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

[ 2 1 1 1 −2 −1 0 3 −5 ][ x y z ]=[ 1 3 2 9 ]

Now, the determinant of A is,

| A |=2( 10+3 )−1( −5+0 )+1( 3−0 ) =26+5+3 =34

Since | A |≠0, thus A is non-singular, therefore, its inverse exists.

Since AX=B, thus, X= A -1 B.

It is known that,

A −1 = adjA | A |

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

A 11 = ( −1 ) 1+1 [ ( −2 )( −5 )−3( −1 ) ] =13

A 12 = ( −1 ) 1+2 [ 1( −5 )−0( −1 ) ] =−( −5 ) =5

A 13 = ( −1 ) 1+3 [ 1×3−0( −2 ) ] =3

A 21 = ( −1 ) 2+1 [ 1( −5 )−3×1 ] =−( −8 ) =8

A 22 = ( −1 ) 2+2 [ 2( −5 )−0×1 ] =−10

A 23 = ( −1 ) 2+3 [ 2×3−0×1 ] =−6

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

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

A 33 = ( −1 ) 3+3 [ 2( −2 )−1×1 ] =−5

So, the value of adjA is,

adjA=[ A 11 A 21 A 31 A 12 A 22 A 32 A 13 A 23 A 33 ] =[ 13 8 1 5 −10 3 3 −6 −5 ]

Since | A |=34, thus,

A −1 = 1 34 [ 13 8 1 5 −10 3 3 −6 −5 ]

Now,

X= A −1 B [ x y z ]= 1 34 [ 13 8 1 5 −10 3 3 −6 −5 ][ 1 3 2 9 ] [ x y z ]= 1 34 [ 34 17 −51 ]

Thus,

[ x y z ]=[ 1 1 2 −3 2 ]

Hence,

x=1, y= 1 2 and z= −3 2 .