Introduction

The identity matrix is one of the most important objects in linear algebra. If you already know how matrix multiplication works, the identity matrix plays the same role for matrices that the number $1$ plays for ordinary numbers.

This article introduces the identity matrix in a simple, intuitive way.

What Is the Identity Matrix?

• The identity matrix is a special square matrix.
• It has $1$’s on the main diagonal (top-left to bottom-right).
• All other entries are $0$.

Examples:

• The $2 \times 2$ identity matrix: $$I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$
• The $3 \times 3$ identity matrix: $$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Why Is It Called the “Number 1” of Matrices?

For ordinary numbers:

• $1 \cdot x = x$
• $x \cdot 1 = x$

For matrices:

• $I \cdot A = A$
• $A \cdot I = A$

This is why the identity matrix behaves like the number $1$.

Key ideas:

• Multiplying by $I$ does not change a matrix.
• Every square matrix has its own identity matrix of the same size.

How the Identity Matrix Works in Multiplication

When you multiply $I$ by a matrix $A$:

• Each row of $I$ “selects” the corresponding row of $A$.
• Because $I$ has $1$’s on the diagonal and $0$’s elsewhere, nothing gets mixed together.

Example: $$I_2 \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ Example: $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} I_2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

Geometric Meaning

If you think of matrices as transformations (like stretching, rotating, or flipping shapes), then:

• The identity matrix is the “do nothing” transformation.
• It leaves every vector exactly where it is.

Example:

• $I_2 (x, y) = (x, y)$
• No stretching, no rotation, no flipping.

Why the Identity Matrix Matters

• It is essential for defining inverse matrices.
• It helps describe systems of equations.
• It appears in many algorithms in linear algebra.
• It is the foundation for understanding more advanced ideas like eigenvalues.

Think of it as the “home base” of matrix operations.

Calculator

Exercises

1. Write down the $2 \times 2$ identity matrix.

   Solution

   $$I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$
2. Compute $I_2 \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$.

   Solution

   $$I_2 \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$$
3. Compute $\begin{pmatrix} 5 & 0 \\ 0 & 7 \end{pmatrix} I_2$.

   Solution

   $$\begin{pmatrix} 5 & 0 \\ 0 & 7 \end{pmatrix} I_2 = \begin{pmatrix} 5 & 0 \\ 0 & 7 \end{pmatrix}$$
4. True or false: Multiplying any $2 \times 2$ matrix by $I_2$ changes its entries.

   Solution

   
   False. Multiplying by $I_2$ does not change any entries.
5. Write the $3 \times 3$ identity matrix.

   Solution

   $$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
6. Compute $I_3 \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$.

   Solution

   $$I_3 \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$
7. Explain in words why the identity matrix is like the number $1$.

   Solution

   
   Because multiplying by the identity matrix leaves any matrix or vector unchanged, just like multiplying a number by $1$ leaves it unchanged.