Answer to Question #109482 in Linear Algebra for Caylin

Question

Let

1) a11 x1 + a12 x2 + a13 x3 = b1
2) a21 x1 + a22 x 2+ a23 x3 = b2
3) a31 x1 + a32 x 2+ a33 x 3 = b3

SHOW that if det(A) does not equal 0, where det (A) is the determinant of the coefficient matrix, then x2= det(A2)/det(A)

where det (A2) is the determinant obtained by replacing the second column of det (A) by (b1, b2, b3) to the power T.

Please assist.

Answer 1

Using Laplace expansion, let's rewrite $\det (A)=C_1a_{12}+C_2a_{22}+C_3a_{32}$ , where $C_i$ are the cofactors, depending on the column of A other than column 2.

Multiplying 1st equation of the system by $C_1$ , 2nd by $C_2$ and 3rd by $C_3$ we get

$x_2(C_1a_{12}+C_2a_{22}+C_3a_{32})=x_2\det (A)=C_1b_1+C_2b_2+C_3b_3$

By the construction of it, the right hand part of this equation is a determinant of A2, which means

$x_2=\frac{\det(A_2)}{\det(A)}$ if $\det(A)\ne0$