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.