User:Djwinters/Cramer's rule 3

Given n linear equations with n variables $x_{1}$ , $x_{2}$ , $\ldots$ , $x_{n}$ .

{\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\ldots +a_{1n}x_{n}&=&b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\ldots +a_{2n}x_{n}&=&b_{2}\\\vdots &\vdots &\vdots \\a_{n1}x_{1}+a_{n2}x_{2}+\ldots +a_{nn}x_{n}&=&b_{n}\end{matrix}}

Cramer's rule gives the solution:

x_{1}={\frac {det\left|{\begin{matrix}b_{1}&a_{12}&\ldots &a_{1n}\\b_{2}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{n}&a_{n2}&\ldots &a_{nn}\end{matrix}}\right|}{det\left|{\begin{matrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{matrix}}\right|}}\ \ x_{2}={\frac {det\left|{\begin{matrix}a_{11}&b_{1}&\ldots &a_{1n}\\a_{21}&b_{2}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&b_{n}&\ldots &a_{nn}\end{matrix}}\right|}{det\left|{\begin{matrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{matrix}}\right|}}\ \ \ldots \ \ x_{n}={\frac {det\left|{\begin{matrix}a_{11}&a_{12}&\ldots &b_{n}\\a_{21}&a_{22}&\ldots &b_{n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &b_{n}\end{matrix}}\right|}{det\left|{\begin{matrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{matrix}}\right|}}

These expressions for $x_{m}$ can be put into matrix notation as follows. First do a Laplace expansion (aka cofactor expansion) on the determinants which are in the numerators using the columns which contain $b_{1}$ , $b_{2}$ , $\ldots$ , $b_{n}$ . Thus Cramer's rule becomes;

$x_{m}={\frac {\begin{matrix}c_{1m}b_{1}+c_{2m}b_{2}+\ldots +c_{nm}b_{n}\end{matrix}}{det\left|{\begin{matrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{matrix}}\right|}}$

Where $c_{rc}$ are the cofactors of the coefficient matrix [A].

$Where\ \ \ \ [A]={\begin{bmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{bmatrix}}\ \ \ [C]={\begin{bmatrix}c_{11}&c_{12}&\ldots &c_{1n}\\c_{21}&c_{22}&\ldots &c_{2n}\\\vdots &\vdots &\ddots &\vdots \\c_{n1}&c_{n2}&\ldots &c_{nn}\end{bmatrix}}\$

$and\ c_{rc}\ =-1^{r+c}M_{rc},\ where\ M_{rc}$ is the determinant of the matrix formed by deleting row r and column c from [A]. Therefore Cramer's rule solutions for $x_{m}$ has the matrix form

${\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}\ ={\frac {{\begin{bmatrix}c_{11}&c_{12}&\ldots &c_{1n}\\c_{21}&c_{22}&\ldots &c_{2n}\\\vdots &\vdots &\ddots &\vdots \\c_{n1}&c_{n2}&\ldots &c_{nn}\end{bmatrix}}^{T}{\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{bmatrix}}}{det\left|{\begin{matrix}a_{11}&a_{12}&\ldots &{a_{1}n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{matrix}}\right|}}\$

$[C]^{T}$ is called the adjugate matrix of $[A]$ , written as adj[A].

Both $[C]^{T}[A]\ and\ [A][C]^{T}$ are equal to det[A] times the idehtity matrix as shown below.

Let\ \ {\begin{bmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{bmatrix}}{\begin{bmatrix}c_{11}&c_{12}&\ldots &c_{1n}\\c_{21}&c_{22}&\ldots &c_{2n}\\\vdots &\vdots &\ddots &\vdots \\c_{n1}&c_{n2}&\ldots &c_{nn}\end{bmatrix}}^{T}={\begin{bmatrix}d_{11}&d_{12}&\ldots &d_{1n}\\d_{21}&d_{22}&\ldots &d_{2n}\\\vdots &\vdots &\ddots &\vdots \\d_{n1}&d_{n2}&\ldots &d_{nn}\end{bmatrix}}

Consider $d_{ij}\ =\ {\begin{matrix}a_{i1}c_{j1}+a_{i2}c_{j2}+\ldots +a_{in}c_{jn}\end{matrix}}$ . When i=j this is just det[A] expressed as the cofactor expansion along row=i. When i not= j this is just the cofactor expansion of the determinant of [A] after row j has been replaced with row i, which is zero since 2 rows are identical.

Similarly

Let\ \ {\begin{bmatrix}c_{11}&c_{12}&\ldots &c_{1n}\\c_{21}&c_{22}&\ldots &c_{2n}\\\vdots &\vdots &\ddots &\vdots \\c_{n1}&c_{n2}&\ldots &c_{nn}\end{bmatrix}}^{T}{\begin{bmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{bmatrix}}={\begin{bmatrix}d_{11}&d_{12}&\ldots &d_{1n}\\d_{21}&d_{22}&\ldots &d_{2n}\\\vdots &\vdots &\ddots &\vdots \\d_{n1}&d_{n2}&\ldots &d_{nn}\end{bmatrix}}

Consider $d_{ij}\ =\ {\begin{matrix}c_{1i}a_{1j}+c_{2i}a_{2j}+\ldots +c_{ni}a_{nj}\end{matrix}}$ . When i=j this is just det[A] expressed as the cofactor expansion along column=j. When i not= j this is just the cofactor expansion of the determinant of [A] after column i has been replaced with column j, which is zero since 2 columns are identical.