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प्रश्न
If A = `[[1,1,1],[0,1,3],[1,-2,1]]` , find A-1Hence, solve the system of equations:
x +y + z = 6
y + 3z = 11
and x -2y +z = 0
उत्तर
A = `[[1,1,1],[0,1,3],[1,-2,1]]`
A11 = 7 , A12 = 3, A13 = -1
A21 = -3 , A22 = 0, A23 = +3
A31 = 2, A32 = -3, A33 = 1
|A| = 1(7) + 3 - 1= 9
`∴ A^(-1) = 1/|A| adj A`
` = 1/9[[7,-3,2],[3,0,-3],[-1,3,1]]`
Verification
AA-1 = I
`= 1/9[[1,1,1],[0,1,3],[1,-2,1]] xx [[7,-3,2],[3,0,-3],[-1,3,1]] `
`= 1/9[[9,0,0],[0,9,0],[0,0,9]]`
=I3
X +Y + Z = 6
0X + Y + 3Z = 11
X -2Y + Z = 0
`[[1,1,1],[0,1,3],[1,-2,1]] [ [X],[Y],[Z]] =[[6],[11],[0]]`
aX =b ⇒ x = A-1 b
`A^(-1) = 1/9 [[7,-3,2],[3,0,-3],[-1,3,1]] `
`∴ [[x],[y],[x]] =A^(-1)b`
`= 1/9 [[7,-3,2],[3,0,-3],[-1,3,1]] [[6],[11],[0]]`
`=1/9 [[42-33],[18],[-6+33]] =1/9 [[9],[18],[27]]`
`=[[1],[2],[3]]`
∴ x =1; y =2; z = 3
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