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Maths > Determinants > Solution of System of Linear Equations using Inverse of a Matrix
Determinants

Solution of System of Linear Equations using Inverse of a Matrix

Equation Solver: Suppose you have bought two types of ice creams for Rs. 500. Suppose your friend asks you the cost of each type of the ice cream. You tell him that the cost of type 1 is 100 more than that of type 2. Your friend can easily calculate the cost of each ice cream. But what if you buy five different varieties of ice creams of different prices for five days continuously?

Also, assume that their prices vary each day by some amount with respect to each other. Isn’t it calculating the prices for each variety time-consuming and confusing? In this section, we will learn how to calculate the solution of a system of linear equations. We will learn how to use determinants and matrices as an equation solver.

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Determinants

A determinant of a square matrix [aij] of order n where aij = (i, j) th element of A is a number (real or complex) associated with it. The number of rows is the same as the number of the columns in a square matrix. If A is a square matrix such as

Equation Solver

Then determinant of A is |A| = Δ = a11 [(a22 × a33) – (a23 × a32)] – a12 [(a21 × a33) – (a31 × a23)] + a13 [(a21 × a32) – (a31 × a22)].


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Determinants and Matrices as Equation Solver

Since now we are familiar with the way of calculating the determinant of a square matrix. Here, we will discuss the way to solve a system of linear equations in two or three variables. With the help of the determinant, we can also check for the consistency of linear equations.

  • Consistent System: If one or more solution(s) exists for a system of equations then it is a consistent system
  • Inconsistent System: A system of equations with no solution is an inconsistent system.

The Solution of System of Linear Equations

A solution for a system of linear Equations can be found by using the inverse of a matrix. Suppose we have the following system of equations

  • a11 x + a12 y + a13 z = b1
  • a21 x + a22 y + a23 z = b2
  • a31 x + a32 y + a33 z = b3

where, x, y, and z are the variables and a11, a12, … , a33 are the respective coefficients of the variables and b1, b2, and b3 are the constants. We need to find the solution for the values of the variables in this system of equations.

Determinant as an Equation Solver

The above system of equations can be represented in the form of a square matrix as

Equation Solver

i.e., AX = B or,

Equation Solver

Here arise two cases

Case1

If A is a non-singular matrix i.e., |A| ≠ 0, then its inverse exists.
We have A X = B
or, A– 1 (A X) = A– 1 B (pre-multiplying by A– 1)
or, (A– 1 A) X = A– 1 B
and, I X  = A– 1 B (I is the identity matrix)
or, X = A– 1 B where,  A– 1 = (adj A) ⁄ |A|
This matrix equation provides a unique solution and is known as the Matrix Method.

Case2

If A is a singular matrix, then |A| = 0 then we calculate (adj A) B. If (adj A) B ≠ 0 (zero matrix), then the solution does not exist. The system of equations is inconsistent. Else, if (adj A) B = 0 then the system will either have infinitely many solutions (consistent system) or no solution (inconsistent system).

Solved Example for You

Question 1: Suppose you have three numbers. The sum of the two numbers and the twice of the second equals 2. The sum of the second and third when subtracted from the twice of first gives 1. The difference of thrice of first and five times the third gives 5. Rewrite the statement in form of the system of equations. Solve it using Matrix Method as an equation solver.

Answer : Assume that x, y, and z are the three numbers. Rewriting the above statement we have the following system of equations

x + 2y + z = 2
2x – y – z = 1
3x – 5y = 5

In matrix notation, we have

Equation Solver

Here, the determinant of A = |A| = 1(5 – 0) – 2(–10 + 3) + 1(0 + 3) = 22 ≠ 0. Hence there exists a unique solution for X.

Calculating adj (A), we have Aij = (–1)(i + j) Mij , where Mij is the co-factor of aij

  • A11 = 1(5 – 0) = 5, A12 = –1(–10 + 3) = 7, A13 = 1(0 + 3) = 3,
  • A21 = –1(–10 –0) = 10, A22 = 1(–5 – 3) = –8, A23 = –1(0 – 6) = 6,
  • A31 = 1(–2 + 1) = –1, A32 = –1(–1 – 2) = 3, A33 = 1(–1 – 4) = –5

Equation Solver

The inverse of the matrix A is A−1.

Equation Solver

Since X = A– 1 B

Equation Solver

Equation Solver

Thus, x = 15⁄22, y = 21⁄22, and z = –13⁄22.

Question 2: What do you mean by consistent and inconsistent system?

Answer: Consistent System refers to when one or more solutions are present for a system of equations. On the other hand, an Inconsistent System is a system of equations having no solutions.

Question 3: What are inverse matrices used for?

Answer: The application of inverse typically is present in structural analysis, where a matrix will represent the properties of a piece of your design. Further, there is a matrix that corresponds to its physical properties and we make use of the inverse to solve the equation or system for strength variables.

Question 4: What is the formula for linear equations?

Answer: The standard form for linear equations in two variables is Ax + By= C. For instance, 3x+4y=7 is a linear equation in standard form. Thus, when you get an equation in this form, it’s quite easy to find both intercepts (x and y).

Question 5: What is a linear and non-linear equation?

Answer: Linear refers to something related to a line. We use all the linear equations to define or construct a line. On the other hand, a non-linear equation is one that does not create a straight line. It resembles a curve in a graph and has got a variable slope value.

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jannat
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jannat

Is determinant available just for square matrix?

Stephen
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Stephen

Yes, it is only defined for square matrices.

J. A. Zahálka
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J. A. Zahálka

It actually is. It is equal to zero for all non-square matrices.

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