# Applications of Determinants and Matrices

Applications of determinants and matrices can be widely seen while checking the consistency of the system of linear equations in two or three variables. We can solve the linear equations in two or three variables using determinants and matrices.

Before we discuss the applications of determinants and matrices to find the solution of linear equations and checking their consistency, let us learn about consistent and inconsistent systems.

Also, read: Determinant To Find Area Of A Triangle

## Consistency of System of Equations

• A system of equations is called consistent if it has one or more solution
• A system of equations is called inconsistent if it does not have a solution

### Solution of system of linear equations using inverse of a matrix

Suppose the system of equations is given by:

a1 x + b1 y + c1 z = d1

a2 x + b2 y + c2 z = d2

a3 x + b3 y + c3 z = d3

Now let us say, A, B and X are three matrices, such that;

$$\mathrm{A}=\left[\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right], \mathrm{X}=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \text { and } \mathrm{B}=\left[\begin{array}{l} d_{1} \\ d_{2} \\ d_{3} \end{array}\right]$$

Hence, the system of equations is given by:

AX = B

$$\left[\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right] =\left[\begin{array}{l} d_{1} \\ d_{2} \\ d_{3} \end{array}\right]$$

Condition 1:

If A is a non-singular matrix, then X = A-1B. The equation gives a unique solution because the inverse of the matrix is unique. Also, such a method of finding the solution for a system of linear equations is called the Matrix method.

Condition 2:

If A is a singular matrix, then determinant of A, |A| = 0.

Now for such a condition, there exist two cases based on (adj A) B.

• If (adj A) B is not equal to the zero matrix, then the system of equations does not have a solution and hence is called inconsistent.
• If (adj A) B is equal to the zero matrix, then the system of equations will have either no solution or infinitely many solutions. Hence, they may be either consistent or inconsistent.

## Solved Examples

Problem 1: Find if the given system of equations is consistent or inconsistent.

x+3y = 5 and 2x + 6y = 8

Solution: Given, the system of equations are:

x+3y = 5 and 2x + 6y = 8

As per the matrix equation, we know;

AX = B

Hence, the system of equations can be written as:

$$\left[\begin{array}{lll} 1 & 3 \\ 2 & 6 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right] =\left[\begin{array}{l} 5\\ 8 \end{array}\right]$$

By determinant formula, we know;

$$|A| = \begin{vmatrix} 1 & 3\\ 2 & 6 \end{vmatrix}$$

|A| = 6 – 6 = 0

Now, the adjoint of matrix A, will be;

$$adj. ~A = \begin{vmatrix} 6 & -2\\ -3 & 1 \end{vmatrix}$$ $$(adj. ~A)B = \begin{bmatrix} 6 & -2\\ -3 & 1 \end{bmatrix} \begin{bmatrix} 5\\ 8 \end{bmatrix} = \begin{bmatrix} 6 \\ -2 \end{bmatrix}$$

Thus, (adj.A)B ≠ 0

Hence, the given system of equations is inconsistent.

Problem 2: If 5x – y + 4z = 5, 2x + 3y + 5z = 2 and 5x – 2y + 6z = –1 are the system of equations, then find if it is consistent or not.

Solution: Given, the system of equations are:

5x – y + 4z = 5

2x + 3y + 5z = 2

5x – 2y + 6z = –1

As per the matrix equation, AX = B, we can write the above system of equations as:

$$\begin{bmatrix} 5 &-1 &4 \\ 2 & 3 & 5\\ 5 & -2 & 6 \end{bmatrix}\begin{bmatrix} x\\ y\\ z \end{bmatrix}= \begin{bmatrix} 5\\ 2\\ -1 \end{bmatrix}$$

If we compare,

$$A = \begin{bmatrix}5 &-1 &4 \\2 & 3 & 5\\ 5 & -2 & 6\end{bmatrix}$$

The determinant of matrix A will be:

$$A = \begin{vmatrix}5 &-1 &4 \\2 & 3 & 5\\ 5 & -2 & 6\end{vmatrix}$$

|A| = 5(18 + 10) + 1(12 -25)+4(-4-15)

= 140 – 13 – 76

= 140 – 89

= 51

So, |A| ≠ 0

Hence, the system of equations is consistent.

## Practice Questions

Find if the given system of equations are consistent or inconsistent.

1. 5x + 2y = 4 and 7x + 3y = 5
2. 2x + 3y +3 z = 5, x – 2y + z = – 4 and 3x – y – 2z = 3

## Frequently Asked Questions on Application of determinant and matrices

### What is the application of determinants?

A determinant is used to find the solution of a system of equations. It is also used to determine if a matrix has an inverse.

### What is the use of determinants in geometry?

Determinants are used to find the area of triangles, when the vertices are known to us.

### What are the applications of matrices?

Matrices is a branch of mathematics that deals with various arithmetical problems in different fields such as mechanics, electrodynamics, optics,etc. It is widely used in scientific fields.

### What is the condition for a consistent system of equations?

If the system of equations has one or more solutions, then it is called consistent.

### What is the condition for an inconsistent system of equations?

If the system of equations does not have a solution, then it is called inconsistent.