Differential Equations and Linear Algebra, 4th Edition

Inverses and the Adjoint Matrix

We now use Cramer’s rule to develop an explicit formula for the inverse ${\mathbf{\text{A}}}^{-1}$ of the invertible matrix A. First, we need to rewrite Cramer’s rule more concisely. Expansion of the determinant in the numerator in Eq. (24) along its i th column yields

$$x_i=\frac{1}{\left|\mathbf{\text{A}}\right|}(b_1{A}_{1i}+b_2{A}_{2i}+\cdots +b_n{A}_{ni}),$$ (25)

because the cofactor of $b_p$ is simply the cofactor $A_{pi}$ of $a_{pi}$ in $\left|\mathbf{A}\right|.$ The formula in Eq. (25) gives the solution vector

$$\mathbf{\text{x}}=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]=\frac{1}{\left|\mathbf{\text{A}}\right|}\left[\begin{array}{ccccccc}{b}_1{A}_{11}& +& b_2{A}_{21}& +& \cdots & +& b_n{A}_{n1}\\ b_1{A}_{12}& +& b_2{A}_{22}& +& \cdots & +& b_n{A}_{n2}\\ ⋮\\ b_1{A}_{1n}& +& b_2{A}_{2n}& +& \cdots & +& b_n{A}_{nn}\end{array}\right].$$

Then the definition of matrix multiplication yields

$$\mathbf{\text{x}}=\frac{1}{\left|\mathbf{\text{A}}\right|}\left[\begin{array}{cccc}{A}_{11}& A_{21}& \cdots & A_{n1}\\ A_{12}& A_{22}& \cdots & A_{n2}\\ ⋮& ⋮& \ddots & ⋮\\ A_{1n}& A_{2n}& \cdots & A_{nn}\end{array}\right]\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right]$$ (26)

for the solution x of $\mathbf{\text{A}}\mathbf{\text{x}}=\mathbf{\text{b}}.$

Observe that the double subscripts in (26) are reversed from their usual order; the element in the ith ...