Theorem – Cramer’s Rule – Math-reference.net

Contents
1. Theorem
2. Proof

Theorem

Let $A$ be a matrix with dimension $n\times n$ , and $\mathbf{x}$ , $\mathbf{x}$ be vectors. Then, consider the system of linear equations given by

$A\mathbf{x} = \mathbf{b}$

Let $a_{ij}$ represent the elements of $A$ , and similarly $x_i$ , $b_i$ represent the elements of $\mathbf{x}$ and $\mathbf{b}$ . Finally, let $A_i$ be the matrix $A$ , except where column $i$ (which will be a column vector) of the matrix $A$ is replaced with the vector $\mathbf{b}$ .

Then,

$x_i = \frac{det(A_i)}{det(A)}$

2. Proof

Let the columns of $A$ be denoted by $c_i$ for $i=1,..., n$ . Then, we can write

$A\mathbf{x} = \mathbf{b} = c_1x_1 + c_2x_2 + ... + c_n x_n$

The key point here is that $\mathbf{b} = c_1x_1 + c_2x_2 + ... + c_n x_n$ . Furthermore, we can write the matrix $A$ as $A = c_1+ c_2 + ... + c_n$ . Then, the matrix $A_i$ , where $c_i$ is replaced with the vector $\mathbf{b}$ , can be written as

$A_i = \mathbf{b} + c_2 + ... + c_n$

If $i=1$ ,

$A_i = c_1 + ... + \mathbf{b} + c_n$

if $i = n$ , or

$A_i = c_1 + ... + \mathbf{b} + ... + c_n$

the determinant of the matrix $A_i$ is therefore of the form above, ie for $i=1$ , $det(A_i) = det(\mathbf{b}~c_2~...~c_n)$ . But, recall that

$\mathbf{b} = c_1x_1 + c_2x_2 + ... + c_n x_n$

So, substituting $\mathbf{b}$ back into $A_i$ , we end up with (in the case that $i=1$ ),

$det(A_i) = det(c_1x_1 + c_2x_2 + ... + c_n ~|~ x_n ~|~ c_2~|~...~|~c_n)$

But, from the laws of determinants, we have that adding multiples of columns together does not change the value of the determinant, and that multiplying one column by a constant, say $x_i$ , multiplies the determinant by $x_i$ . Thus,

$det(A_i) = x_i \times det(A)$

Which, by the assumption that $A$ is not singular (and hence $det(A)\neq 0$ , we have

$x_i = \frac{det(A_i)}{det(A)}$

And this proves the theorem. $\quad \square$

NB – The inverse of a matrix is a special use of cramer’s rule, where $\mathbf{b}$ is the column of zeros and a one for each column of the identity matrix.