Linear Algebra/Cramer's Rule

Cramer's Rule

Let's try to solve the systems of linear equations:
a₁₁x₁+a₁₂x₂+a₁₃x₃+...+a_1nx_n=b₁
a₂₁x₁+a₂₂x₂+a₂₃x₃+...+a_2nx_n=b₂
a₃₁x₁+a₃₂x₂+a₃₃x₃+...+a_3nx_n=b₃
...
a_n1x₁+a_n2x₂+a_n3x₃+...+a_nnx_n=b_n

Which is the special case when the number of equations and the number of variables are the same.

Consider the matrix

$\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & a_{2n}\\ a_{31}& a_{32}& a_{33}& \dots & a_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n1}& a_{n2}& a_{n3}& \dots & a_{nn}\end{array}\right]$

to be denoted D.

First, we multiply the nth equation by the cofactor Co(a_nj) for the jth column, and add it all up. This gets us

Co(a_1j)a₁₁x₁+Co(a_1j)a₁₂x₂+Co(a_1j)a₁₃x₃+...+Co(a_1j)a_1nx_n+
Co(a_2j)a₂₁x₁+Co(a_2j)a₂₂x₂+Co(a_2j)a₂₃x₃+...+Co(a_2j)a_2nx_n+
Co(a_3j)a₃₁x₁+Co(a_3j)a₃₂x₂+Co(a_3j)a₃₃x₃+...+Co(a_3j)a_3nx_n+
+...+
Co(a_nj)a_n1x₁+Co(a_nj)a_n2x₂+Co(a_nj)a_n3x₃+...+Co(a_nj)a_nnx_n
=
Co(a_1j)b₁+Co(a_2j)b₂+Co(a_3j)b₃+...+Co(a_nj)b_n.

The left side cancels out except for Co(a_1j)a_1jx_j+Co(a_2j)a_2jx_j+Co(a_3j)a_3jx_j+...+Co(a_nj)a_njx_j

which is equal to $x_j\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & a_{2n}\\ a_{31}& a_{32}& a_{33}& \dots & a_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n1}& a_{n2}& a_{n3}& \dots & a_{nn}\end{array}\right]=D$

and the right side is equal to

$\left[\begin{array}{ccccccc}{a}_{11}& a_{12}& a_{13}& \dots & b_1& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & b_2& \dots & a_{2n}\\ a_{31}& a_{32}& a_{33}& \dots & b_3& \dots & a_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n1}& a_{n2}& a_{n3}& \dots & b_n& \dots & a_{nn}\end{array}\right]$, to be denoted D(j), which is the same thing as D but with the jth column replaced by b_k.

Dividing by D gets x_j=$\frac{D_j}{D}$.

This formula is called Cramer's Rule, and this solution exists when D is not equal to 0.

In particular, in the process of finding the solution, we also find that this is the only solution, so this solution is unique.

Example

Consider the system of linear equations below.
$3x_1+2x_2-5x_3=15$
$5x_1+x_3=23$
$x_2+x_3=12$

If we only want the solution for, say, $x_2$, we can apply Cramer's Rule to find that its solution is $\frac{D_2}{D}$, and since we know
$D_2=\left[\begin{array}{ccc}3& 15& -5\\ 5& 23& 1\\ 0& 12& 1\end{array}\right]$,
$x_2=\frac{\det \left[\begin{array}{ccc}3& 15& -5\\ 5& 23& 1\\ 0& 12& 1\end{array}\right]}{\det \left[\begin{array}{ccc}3& 2& -5\\ 5& 0& 1\\ 0& 1& 1\end{array}\right]}=9$.