CBSE Class 12-science Answered

Simultaneous linear equations in three variable are

3 x - 2 y + 3 z = 8

2 x + y - z = 1

4x - 3y + 2z = 4

Above equations are written using matrix A as

A Ψ = φ

where Ψ and φ are column vectors

To get the colum vector Ψ = ( x  y  z )^T , we need to find inverse of matrix A , i.e. A^-1 .

First step to get inverse of matrix A is to get cofactor matrix

Let us denote a_ij is element of i-th row anf j-th column of matrix A .

Cofactor of a_ij is value of derminant obtained after leaving t-th row and j-th colum element with sign (-1)^i+j

For example cofactor of a₁₁ = 3 is calculated as follows

cofactor of a₁₂ = -8 ;  cofactor of a₁₃ = -10 ;

cofactor of a₂₁ = -5 ;  cofactor of a₂₂ = -6 ; cofactor of a₂₃ = 1 ;

cofactor of a₃₁ = -1 ;  cofactor of a₃₂ = 9 ; cofactor of a₃₃ = 7 ;

Cofactor matrix of matrix A is

Adjoint matrix A⁺ of matrix A is transpose of cofactor matrix

Inverse matrix A^-1 is  Adjoint matrix A⁺ divided by determinant of matrix A

Determinant of A is calculated as given below

Hence inverse matrix is

Colum vector ψ = ( x y z )^T is calculated by multilying colum vector φ = ( 8  1   4 )^T by inverse matrix

Hence we get ,  x = 1 ; y =2 ; z = 3