Question

What is Cramer's rule in determinants?

Answer 1

Cramer's rule:

• The solution obtained using Cramer’s rule are the determinants of the coefficient matrix and determinants obtained from it by replacing one column with the column vector of the right-hand sides of the equations.
• It is also known as the determinant method.
• It is used to find the solution to a system of linear equations with as many equations.

Example: The Cramer's rule for a $3\times 3$

Consider a system of three linear equations with variables $x,y$ and $z$.

$a_{1}x+b_{1}y+c_{1}z=d_1$

$a_{2}x+b_{2}y+c_{2}z=d_2$

$a_{3}x+b_{3}y+c_{3}z=d_3$

Step $1$: Rewrite the system in the matrix form $AX=B$

The coefficient matrix is: $A=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$

The variable matrix is: $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

The constant matrix is: $B=\left[\begin{array}{c}{d}_1\\ d_2\\ d_3\end{array}\right]$

So, the matrix form of the system is,

$\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}{d}_1\\ d_2\\ d_3\end{array}\right]$

Step $2$: calculate the determinants $D,D_x,D_y$ and $D_z$

The determinant of the coefficient matrix $A$ is,

$D=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|\begin{array}{}\\ \\ \end{array}$

$D_x$: The first column of $D$ is replaced by the elements of the constant matrix.

$D_x=\left|\begin{array}{ccc}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b3& c_3\end{array}\right|\begin{array}{}\\ \\ \end{array}$

$D_y$: The second column of $D$ is replaced by the elements of the constant matrix.

$D_y=\left|\begin{array}{ccc}{a}_1& d_1& c_1\\ a_2& d_2& c_2\\ a_3& d_3& c_3\end{array}\right|\begin{array}{}\\ \\ \end{array}$

$D_z$: The third column of $D$ is replaced by the elements of the constant matrix.

$D_z=\left|\begin{array}{ccc}{a}_1& b_1& d_1\\ a_2& b_2& d_2\\ a_3& b_3& d_3\end{array}\right|\begin{array}{}\\ \\ \end{array}$

Step $3$: Finding the values of the variables

The values of the variables $x,y$ and $z$, when $D\ne 0$ are:

$x=\frac{D_x}{D}$

$y=\frac{D_y}{D}$

$z=\frac{D_z}{D}$