Cramer’s Rule

Cramer’s Rule is a mathematical theorem used to solve linear systems of equations with as many equations as unknowns, provided that the system has a unique solution. It expresses the solution in terms of the determinants of matrices.

Statement

Consider a system of $n$ linear equations with $n$ unknowns, which can be represented using the matrix equation.

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

where:

• $A$ is an $n\times n$ coefficient matrix.
• $\vec{x}$ is the column vector of unknowns.
• $\vec{b}$ is the column vector of constants.

If the determinant of the coefficient matrix $A$ is non-zero ($\det (A)\ne 0$), then the system has a unique solution given by:

$$x_i=\frac{\det (A_i)}{\det (A)}\text{for }i=1,2,\dots ,n$$

where $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with the vector $\vec{b}$.

Example

Consider the following system of linear equations:

$$\{\begin{array}{l}3x_1-2x_2=6\\ -5x_1+4x_2=8\end{array}$$

We can represent this system in matrix form:

$$A=\left[\begin{array}{cc}3& -2\\ -5& 4\end{array}\right],\vec{x}=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right],\vec{b}=\left[\begin{array}{c}6\\ 8\end{array}\right]$$

Calculating the determinant of $A$:

$$\det (A)=(3)(4)-(-2)(-5)=12-10=2\ne 0$$

Now, we compute the matrices $A_1$ and $A_2$:

$$A_1=\left[\begin{array}{cc}6& -2\\ 8& 4\end{array}\right],A_2=\left[\begin{array}{cc}3& 6\\ -5& 8\end{array}\right]$$

Calculating their determinants:

$$\det (A_1)=(6)(4)-(-2)(8)=24+16=40$$ $$\det (A_2)=(3)(8)-(6)(-5)=24+30=54$$

Finally, we can find the solutions for $x_1$ and $x_2$ using Cramer’s Rule:

$$x_1=\frac{\det (A_1)}{\det (A)}=\frac{40}{2}=20x_2=\frac{\det (A_2)}{\det (A)}=\frac{54}{2}=27$$