# The Identity Matrix

# The Identity Matrix

## Definition and Characteristics

- The
**Identity Matrix**, denoted by**I**, is a special type of matrix which has ones on its main diagonal and zeros elsewhere. - It is termed “identity” due to its property that when any matrix is multiplied by it, the original matrix is returned.
- This concept is analogous to multiplying numbers by one, where the original number is unchanged.

## Types of Identity Matrices

**Square Identity Matrix**: An identity matrix is always a**square matrix**, meaning it has the same number of rows as columns.- Examples of square identity matrices include 2x2, 3x3, 4x4, and so on.

## Identity Matrix and Matrix Multiplication

- When an identity matrix is used in matrix multiplication, it essentially has no effect.
- In other words, for any matrix
**A**,**AI = IA = A**would hold true, where**A**is any matrix,**I**is the identity matrix, and**AI**and**IA**are the results of multiplying the matrix**A**by**I**from the right and the left respectively. - This rule applies irrespective of the order in which the matrix and identity matrix are multiplied.

## Finding the Identity Matrix

- The identity matrix of a given matrix can be found by performing row operations until the matrix is in
**Row Echelon form**, a form with ones down the leading diagonal and zeros everywhere else.

## Final Note

- Understanding the properties and application of identity matrix is crucial for solving complex matrix-related problems. Regular practise can help in better understanding and application of this concept.