Question

2x + 3y +3 z = 5x-2y + z =-43x-y-2z = 313°

Answer 1

The given system of equations is,

2x+3y+3z=5

x−2y+z=−4

3x−y−2z=3

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

[ 2 3 3 1 −2 1 3 −1 −2 ][ x y z ]=[ 5 −4 3 ]

Now, the determinant of A is,

| A |=2( 4+1 )−3( −2−3 )+3( −1+6 ) =10+15+15 =40

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 4+1 =5

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

A 13 = ( −1 ) 1+3 [ −1+6 ] =5

A 21 = ( −1 ) 2+1 [ −6+3 ] =−( −3 ) =3

A 22 = ( −1 ) 2+2 [ −4−9 ] =−13

A 23 = ( −1 ) 2+3 [ −2−9 ] =−( −11 ) =11

A 31 = ( −1 ) 3+1 [ 3+6 ] =9

A 32 = ( −1 ) 3+2 [ 2−3 ] =−( −1 ) =1

A 33 = ( −1 ) 3+3 [ −4−3 ] =−7

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 ] =[ 5 3 9 5 −13 1 5 11 −7 ]

Since | A |=10, thus,

A −1 = 1 40 [ 5 3 9 5 −13 1 5 11 −7 ]

Now,

X= A −1 B [ x y z ]= 1 40 [ 5 3 9 5 −13 1 5 11 −7 ][ 5 −4 3 ] [ x y z ]= 1 40 [ 40 80 −40 ]

Thus,

[ x y z ]=[ 1 2 −1 ]

Hence,

x=1, y=2 and z=−1.