Matrix - Unit 3 | Class 10 | Notes and Important Formulae | Optional Mathematics

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Find the notes here:

1. Introduction To Matrix

2. Types of Matrix

3. Determinant of 2x2 Matrix

4. Inverse of 2x2 Matrix

5. Solution of a System of Linear Equations - Matrix Method

6. Cramer's Rule

Out of Syllabus

7. Transpose of a Matrix

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Overview: Matrix is the way of storing data, especially numbers in rectangular array, arranged in rows and columns format. Matrix allows us to solve the problems of linear equations. These can be solved using determinant, inverse of a matrix and cramer's rule. We will learn briefly about all these here in this notes.

Contents

Determinant

Inverse of a Matrix

Solution of System of Linear Equations

Cramer's Rule

To learn the basics of matrix, follow these links:

Introduction and Types of Matrices

Operation On Matrices and Their Properties

Determinant:

Determinant is defined only for square matrices. In a 2x2 square matrix, it is a number calculated by subtracting the product of the other two elements (in diagonal) from the product of the principal diagonal elements.

Determinant of a matrix is written as |A| or |A+B| or |B|. The name of the matrix should be written inside '||'.

In a 1x1 square matrix, determinant is the same element in the matrix. i.e. A = [a] and |A| = a.

Here is one solved example on determinant of 2x2 matrix:

Click here to learn the explanation of this solution.

Inverse Matrix:

In Inverse Matrix, we need to consider the following topics:

• Singular Matrix
• Non-Singular Matrix and
• Inverse Matrices

A square matrix is said to be a singular matrix if its determinant is equal to 0.

A square matrix is said to be a non-singular matrix if its determinant is not equal to 0.

Remember, Only inverse of a non-singular square matrix can be calculated. Square singular matrix cannot have its inverse matrix.

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Inverse Matrices, when multiplied give the product as Identity Matrix. Hence, two non-singular square matrices, let 'A' and 'B' are said to be inverse matrices if AB = BA = I, and all these matrices have the same order or size.

When A and B are inverse matrices, we can write:

A^-1 = B 

or, B^-1 = A

The inverse matrix of matrix A of following order and elements can be found as follows:

[To be updated] Click here to see the process of finding the inverse of a matrix. 

Solution Of A System Of Linear Equations:

Using the Inverse of Matrices and notating the linear equations in matrix form, we can find the solution of the system of linear equation.

A linear equation is that equation which can be written in the form of ax+by = c.

Let us take an example:

3x -3y = 6

5x +6y = -1

Using multiplication of matrices and relation of equal matrices, this system can be written as follows:

Here, we get,

AX = B

If |A|≠ 0, we can find the inverse of matrix A.

So,

A^-1 (AX) = A^-1 B

or, (A^-1 A)X = A^-1 B

or, I X = A^-1 B

or, X = A^-1 B

Using this identity, we can solve the system linear equations.

[To be updated] Click here to see a solved example of Solution of a system of Linear Equations.

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Cramer's Rule:

Cramer's Rule is used to solve the system of linear equations in two variables using determinants. 

According to this rule,

When we have the following linear equations:

a₁x +b₁y = c₁

And, a₂x + b₂y = c₂

Here, x = ( c₁b₂ -c₂b₁) / (a₁b₂ -a₂b₁)

And, y = ( a₁c₂ - a₂c₁) / ( a₁b₂ -a₂b₁)

While finding x and y, the denominators remain the same but the numerator are different in each cases. 

Let D₁ = c₁b₂ -c₂b₁

And D₂ = a₁c₂ - a₂c₁

Also D = a₁b₂ -a₂b₁

We can write: x = D₁/D and y = D₂/D

[To be updated] Click here to learn about derivation of formula of x and y according to Cramer's rule.

[To be updated] See this example to know step by step explanation of Cramer's rule and solution. 

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