Differential Equations and Linear Algebra, 4th Edition

Cramer’s Rule and Inverse Matrices

Suppose that we need to solve the $n\times n$ linear system

$$\mathbf{\text{A}}\mathbf{\text{x}}=\mathbf{\text{B}},$$ (18)

where

$$\begin{array}{llll}\mathbf{\text{A}}=\left[a_{ij}\right],& \mathbf{\text{x}}=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]& \text{and}& \mathbf{\text{B}}=\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right]\end{array}.$$ (19)

We assume that the coefficient matrix A is invertible, so we know in advance that a unique solution x of (18) exists. The question is how to write x explicitly in terms of the coefficients $a_{ij}$ and the constants $b_i.$ In the following discussion, we think of x as a fixed (though as yet unknown) vector.

If we denote by ${\mathbf{\text{a}}}_1,{\mathbf{\text{a}}}_2,\dots ,{\mathbf{\text{a}}}_n$ the column vectors of the $n\times n$ matrix A, then

$$\mathbf{\text{A}}=\left[\begin{array}{llll}{\mathbf{\text{a}}}_1& {\mathbf{\text{a}}}_2& \cdots & {\mathbf{\text{a}}}_n\end{array}\right].$$ (20)

By Fact 1 in Section 3.5, we can rewrite Eq. (18) as

$$x_1{\mathbf{\text{a}}}_1+x_2{\mathbf{\text{a}}}_2+\cdots x_n{\mathbf{\text{a}}}_n=\mathbf{\text{b}}.$$

Thus the constant vector b is expressed in terms of the entries $x_1,x_2,\dots ,x_n$ of the solution vector x and the column vectors of A by

$$\mathbf{\text{b}}={\displaystyle \sum _{j=1}^n{x}_j{\mathbf{\text{a}}}_j}$$