Question

Use the method of solving simultaneous equations by finding the inverse of the matrix of coefficients, together with the formula ${\mathbf{A}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{d}\mathbf{e}\mathbf{t}\mathbf{}\mathbf{A}}{\mathbf{C}}^{\mathbf{T}}$for the inverse of a matrix, to obtain Cramer’s rule.

Short answer

The Cramer’s rule is proved by using the equation ${\mathrm{A}}^{-1}\mathrm{b}=\mathrm{x}$.

Step-by-step solution

Definition of inverse of a matrix:

The inverse of a matrix A is defined as the matrix ${\mathbf{A}}^{\mathbf{-}\mathbf{1}}$such that ${\mathbf{AA}}^{\mathbf{-}\mathbf{1}}$and ${\mathbf{A}}^{\mathbf{-}\mathbf{1}}\mathbf{A}$are both equal to a unit matrix l .

 Prove Cramer’s rule:

The Cramer’s rule is to be proved by the use of the method of solving simultaneous equations by finding the inverse of the matrix of coefficients, together with the formula ${\mathrm{A}}^{-1}=\frac{1}{\det \mathrm{A}}{\mathrm{C}}^{\mathrm{T}}$for the inverse of a matrix.

Take a $3\times 3$system of linear equations for simplicity.

$$\begin{gathered} a_{11}{x}_1+a_{12}{x}_2+a_{13}{x}_3=b_1 \\ {a}_{21}{x}_1+a_{22}{x}_2+a_{23}{x}_3=b_2 \\ {a}_{31}{x}_1+a_{32}{x}_2+a_{33}{x}_3=b_3 \end{gathered}$$

Write these equations in vector form as Ax = b , where A is the matrix of coefficients.

role="math" localid="1658985824301" $\mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]$

According to Cramer’s rule, the solution of this system is of the form ${\mathrm{x}}_{\mathrm{i}}=\frac{{\det }_{\mathrm{i}}}{\det \mathrm{A}}$, where ${\det }_{\mathrm{i}}$is the determinant of matrix A whose ith column has been replaced by vector b .

If the A matrix is invertible then the solution of the system is $\mathrm{x}={\mathrm{A}}^{-1}\mathrm{b}$.

Apply the formula ${\mathrm{A}}^{-1}=\frac{1}{\det \mathrm{A}}{\mathrm{C}}^{\mathrm{T}}$for the inverse of the matrix.

$\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)=\frac{1}{detA}\left[\begin{array}{ccc}{C}_{11}& C_{21}& C_{31}\\ C_{12}& C_{22}& C_{32}\\ C_{13}& C_{23}& a_{33}\end{array}\right]\left(\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right)$

Therefore,

role="math" localid="1658985892371" $$\begin{gathered} {\mathrm{x}}_1=\frac{1}{\det \mathrm{A}}\left({\mathrm{b}}_1{\mathrm{C}}_{11}+{\mathrm{b}}_2{\mathrm{C}}_{21}+{\mathrm{b}}_3{\mathrm{C}}_{31}\right) \\ =\frac{1}{\det \mathrm{A}}\left({\mathrm{b}}_1\left(+\left|\begin{array}{cc}{\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right|\right)+{\mathrm{b}}_2\left(-\left|\begin{array}{cc}{\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right|\right)+{\mathrm{b}}_3\left(\left|\begin{array}{cc}{\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\end{array}\right|\right)\right) \\ =\frac{1}{\det \mathrm{A}}\left|\begin{array}{ccc}{\mathrm{b}}_1& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{b}}_2& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{b}}_3& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right| \end{gathered}$$

Hence, the Cramer’s rule has been proved.