Cramer’s Rule – Formula , Examples | JEE Maths, CBSE Maths, Maths Honours-1

Learn Cramer’s Rule for JEE Maths, CBSE Maths, B.Sc. Maths, Maths Honors with step-by-step examples, determinants, and applications to solving systems of linear equations. There are two types of finding the solution for a linear system of of equations. One of them is the most powerful Cramer’s Rule and the second method is finding the inverse of a matrix and solving the system of linear equations. We will solve a problem using Cramer’s Rule in this article for clearing the concepts practically.

What is Cramer’s Rule ?

The Cramer’s Rule is the method for solving the linear system of equations using the determinants. When the determinant of the coefficients of the variables is not equal to zero, then the Cramer’s Rule will be applicable. When the determinant of matrix of coefficients is not equal to zero it gives a unique solution. For solving the linear system of equations, the Cramer’s Rule is used as per the below equations.

Let the linear system of equations is as given below:

$\begin{array}{c}\mathrm{px}+\mathrm{qy}=r\\ \mathrm{ax}+\mathrm{by}=c\end{array}$

When expressed as matrix the above can be written as,

\left[{\begin{array}{cc}{p}&{q}\\{a}&{b}\end{array}}\right]\left[{\begin{array}{c}{x}\\{y}\end{array}}\right]\;=\;\left[{\begin{array}{c}{r}\\{c}\end{array}}\right]

In the above equation of matrix the matrix of variables is given by,

\left[{\begin{array}{c}{x}\\{y}\end{array}}\right]

The matrix of constants on right hand side is given by,

\left[{\begin{array}{c}{r}\\{c}\end{array}}\right]

The matrix of coefficients of variables x and y is given as ,

\left[{\begin{array}{cc}{p}&{q}\\{a}&{b}\end{array}}\right]

Determinants of various matrices

The determinant of above coefficient matrix is given by,

\Delta=\left|{\begin{array}{cc}{p}&{q}\\{a}&{b}\end{array}}\right|={pb}-{aq}

The values of determinants of variables x and y are given as ,

{\Delta}_{x}=\left|{\begin{array}{cc}{r}&{q}\\{c}&{b}\end{array}}\right|

{\Delta}_{y}=\left|{\begin{array}{cc}{p}&{r}\\{q}&{c}\end{array}}\right|

Solutions of variables x and y

By using the above matrices determinants, we get the values of variables x and y using the Crmaer’s Rule are given as below.

$x=\frac{{\Delta }_x}{\Delta }$

$y=\frac{{\Delta }_y}{\Delta }$

As above the values of variables x and y are found out using the above equations.The values of various determinants are found out from the equations of the determinants.

Limitations of applicability of Cramer’s Rule

The following are the limitations of applicability of the Cramer’s Rule :

1. The determinant of the coefficients matrix can not be zero, meaning $\Delta \ne 0$
2. At the same time if ${\Delta }_x=0\mathrm{and}\Delta =0$ , then the ratio $\frac{{\Delta }_x}{\Delta }$ is also not definable, as $\frac{0}{0}\mathrm{is}\mathrm{not}\mathrm{definable}$.
3. If the matrix is not singular and $\Delta \ne 0$ has a unique solution.
4. The points no 1 & 2 are true for any variable.

Types of Solutions for system of linear Equations

The various types of solution for the given system of linear equations are given in the following paragraphs.

1. Unique Solution

The linear system of equations will have unique solution when$\Delta \ne 0$ . The system is consistent.

Meaning geometrically the planes in 3D intersect at one unique point.

2. No Solution

When the value of $\Delta =0\mathrm{and}\mathrm{when}\mathrm{atleast}\mathrm{one}{\Delta }_i\ne 0$ , then the system is inconsistent and has no solution.

The planes in 3D do not share a unique intersection point.

3. Infinite or No solution ( Ambiguous Case )

When $\Delta ={\Delta }_x={\Delta }_y={\Delta }_z=0$, then the ratios are indeterminate , here Cramer’s Rule fails to provide the exact behavior of the system of linear equations.

The system is consistent with infinite solutions with infinite solutions or inconsistent with no solution but does not guarantee anything by only using the Cramer’s Rule.

The above topics in detail in the next class where we take up the solutions of system of linear equations using the rank of the matrix. The ranks of matrices also provide us the best insight into the solutions to system of linear equations. Here we confine our study up to Cramer’s Rule.

Homogeneous Equations

The homogeneous equations are the equations where the constant matrix is column matrix with all zeros. It is represented as below.

\left[{\begin{array}{cc}{p}&{q}\\{a}&{b}\end{array}}\right]\left[{\begin{array}{c}{x}\\{y}\end{array}}\right]\;=\;\left[{\begin{array}{c}{0}\\{0}\end{array}}\right]

The discussion on Homogeneous Equations and Rank of Matrix will be continued in next articles.

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A visual image of above Cramer’s Rule discussion is given below image.