Identity Matrices


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Inverse Matrix
More Lessons on Matrices
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In these lessons, we will learn about the identity matrix and inverse matrices.
We also feature a matrix calculator that will help you to find the inverse of a 3×3 matrix. Use it to check your answers.




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The identity matrix and the inverse of a matrix are closely connected.

  1. The Identity Matrix:
  • It’s a square matrix (same number of rows and columns).
  • It has 1s on the main diagonal (from top-left to bottom-right) and 0s everywhere else.
  • It’s denoted by I or In (where n is the size).
  • When you multiply any matrix by the identity matrix (of compatible dimensions), you get the original matrix back. (AI = IA = A)
  1. The Inverse of a Matrix:
  • Only square matrices can have inverses.
  • The inverse of a matrix A, denoted as A-1, is another matrix that, when multiplied by A, gives you the identity matrix I.
    A A-1 = I
    A-1 A = I

The Key Relationship:
The inverse of a matrix “undoes” the effect of the original matrix in matrix multiplication. The identity matrix is the result of this “undoing,” similar to how multiplying a number by its reciprocal (e.g., 5 × 1/5 = 1) gives you 1.

Analogy to Numbers:
Think of it like this:
In regular multiplication, the number 1 is the multiplicative identity because any number multiplied by 1 equals itself (e.g., 7 × 1 = 7).
The reciprocal of a number (its inverse) is what you multiply it by to get 1 (e.g., 7 × 1/7 = 1).

In the same way:
The identity matrix I is the multiplicative identity for matrices.
When you multiply a matrix A with its inverse you will get the identity matrix I.

Identity & Inverse Matrices

Learn how to calculate the inverse of a matrix

A square matrix, I is an identity matrix if the product of I and any square matrix A is A.
i.e. IA = AI = A

For a 2 × 2 matrix, the identity matrix for multiplication is

When we multiply a matrix with the identity matrix, the original matrix is unchanged.


 

 

If the product of two square matrices, P and Q, is the identity matrix then Q is an inverse matrix of P and P is the inverse matrix of Q.
i.e. PQ = QP = I

The inverse matrix of A is denoted by A-1. (read as “A inverse”)

AA-1 = A-1A = I

Note that the inverse of A-1 is A.

Example:

Given that B is the inverse of A, find the values of x and y.

Solution:
AB =

Since B is an inverse of A, we know that AB = I

matrix identity
 

1 – 2y = 1
2y = 0
y = 0

2x = 1
x =




The Identity Matrix
This video introduces the identity matrix and illustrates the properties of the identity matrix.

A n × n square matrix with a main diagonal of 1’s and all other elements 0’s is called the identity matrix In.

If A is a m × n matrix, then ImA = A and AIn = A.
Is A is a n × n square matrix, then
AIn = InA = A.

Determining a 2×2 Inverse Matrix Using a Formula
This video explains the formula used to determine the inverse of a 2x2 matrix, if one exists.

Example:
Determine the inverse of matrix A.

How to find the inverse of a 2×2 matrix using the inverse formula?

Identity matrix and inverse matrix



This matrix calculator will help you find the inverse of a 3×3 matrix.


Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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