Identity Matrices

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 I_n (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)

2. 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.

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

 

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 I_n.

If A is a m × n matrix, then I_mA = A and AI_n = A.
Is A is a n × n square matrix, then
AI_n = I_nA = A.

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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.

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

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Identity matrix and inverse matrix

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

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