1. What Is an Identity Matrix?
An identity matrix is a special type of scalar matrix in which all diagonal entries are 1 and every off-diagonal entry is 0. It is the matrix equivalent of the number 1 in ordinary multiplication.
An identity matrix of order n × n is denoted by \( I_n \).
In general:
\( I_n = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix} \)
2. Understanding the Structure of an Identity Matrix
The identity matrix is always a square matrix. Its key features are:
- The main diagonal contains all 1s.
- Every non-diagonal position contains 0.
- Its form never changes — only its size (order) changes.
2.1. Examples of Identity Matrices
- Identity matrix of order \( 1 \times 1 \):
\( I_1 = [1] \)
- Identity matrix of order \( 2 \times 2 \):
\( I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
- Identity matrix of order \( 3 \times 3 \):
\( I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
2.2. Visual Pattern
The ones run diagonally from the top-left corner to the bottom-right. Everything else is zero. This pattern remains the same no matter how large the matrix becomes.
3. How to Identify an Identity Matrix
A matrix is an identity matrix if and only if:
- It is square.
- Every main diagonal entry is 1.
- Every other entry is 0.
Even a single diagonal entry not equal to 1 makes the matrix not an identity matrix.
4. Why the Identity Matrix Is Important
The identity matrix behaves like the number 1 in matrix multiplication:
\( I_n A = A I_n = A \)
No matter which side it is multiplied on, it leaves the other matrix unchanged.
Some important uses of the identity matrix include:
- Defining the inverse of a matrix.
- Solving systems of linear equations.
- Expressing matrix transformations.
- Building concepts like eigenvalues, diagonalisation, and vector spaces.
Because of this special behaviour, the identity matrix plays a central role throughout linear algebra.