Question

What is Cramer's rule?

Answer 1

Step 1. Cramer's rule:

Cramer's rule is one of the important methods applied to solve a system of equations. In this method, the values of the variables in the system are to be calculated using the determinants of matrices. Thus, Cramer's rule is also known as the determinant method.

Step 2. Example of Cramer's rule:

Consider the equations:

$$\begin{gathered} 3x-2y+z=9 \\ x+2y-2z=-5 \\ x+y-4z=-2 \end{gathered}$$

Then the matrix formed is

$A=\left[\begin{array}{ccc}3& -2& 1\\ 1& 2& -2\\ -2& 1& -4\end{array}\right],b=\left[\begin{array}{c}9\\ -5\\ -2\end{array}\right]$

Now, determinant,

$$\begin{gathered} D=\left|\begin{array}{ccc}3& -2& 1\\ 1& 2& -2\\ 1& 1& -4\end{array}\right| \\ =3\left[-8+2\right]+2\left[-4+2\right]+1\left[1-2\right] \\ =-18-4-1 \\ =-23 \end{gathered}$$

$$\begin{gathered} D_X=\left|\begin{array}{ccc}9& -2& 1\\ -5& 2& -2\\ -2& 1& -4\end{array}\right| \\ =9\left[-8+2\right]+2\left[20-4\right]+1\left[-5+4\right] \\ =-54+32-1 \\ =-23 \end{gathered}$$

$$\begin{gathered} D_Y=\left|\begin{array}{ccc}3& 9& 1\\ 1& -5& -2\\ 1& -2& -4\end{array}\right| \\ =3\left[20-4\right]-9\left[-4+2\right]+1\left[-2+5\right] \\ =48+18+3 \\ =69 \end{gathered}$$

Now,

$$\begin{gathered} x=\frac{D_x}{D} \\ =\frac{-23}{-23} \\ =1 \\ y=\frac{D_y}{D} \\ =\frac{69}{-23} \\ =-3 \\ \therefore x=1,y=-3 \end{gathered}$$

Thus, Cramer's rule will be the best way to find the solution to a system of linear equations with as many equations as unknowns.