The identity matrix Revision Notes for AQA GCSE Further Maths

The identity matrix

What is the identity matrix?

The identity matrix is a very special type of matrix that acts like the number 1 in regular multiplication. In 2×2 form, it has ones along the main diagonal (from top-left to bottom-right) and zeros everywhere else.

The identity matrix is written as I and looks like this:

$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

This might seem simple, but this matrix has some remarkable properties that make it incredibly important in mathematics.

Note

The identity matrix gets its name because it "preserves the identity" of any matrix it's multiplied with - just like how multiplying a number by 1 leaves that number unchanged.

Key properties of the identity matrix

The multiplication property

The most important feature of the identity matrix is how it behaves when you multiply it with other matrices. Just like multiplying any number by 1 gives you the same number back, multiplying any matrix by the identity matrix gives you the original matrix back.

For any matrix A, we have:

$\mathbf{A} \times \mathbf{I} = \mathbf{A}$
$\mathbf{I} \times \mathbf{A} = \mathbf{A}$

This means it doesn't matter whether you multiply the identity matrix on the left or the right - you always get the original matrix back unchanged. This is why it's called the "identity" matrix - it preserves the identity of whatever matrix you multiply it with.

Example

Worked Example: Identity Matrix Multiplication

Let's multiply a matrix by the identity matrix:

$\mathbf{A} = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}, \quad \mathbf{I} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$\mathbf{A} \times \mathbf{I} = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} = \mathbf{A}$

The result is exactly the same as the original matrix!

As a transformation matrix

When the identity matrix is used as a transformation matrix in coordinate geometry, something very interesting happens - absolutely nothing! Points remain exactly where they are. There is no movement, no rotation, no reflexion, and no scaling. Every point stays in its original position.

This makes the identity matrix the "do nothing" transformation, which is actually very useful as a reference point when studying other transformations.

Note

The identity transformation serves as a baseline - it's like having a "zero change" reference that helps us understand what other transformations actually do to shapes and coordinates.

Understanding the structure

The identity matrix follows a clear pattern:

The main diagonal (top-left to bottom-right) contains only 1s
All other positions contain 0s
For a 2×2 matrix, this gives us the form shown above

This structure is what gives the identity matrix its special properties. The 1s on the diagonal ensure that each component of a vector or matrix stays the same when multiplied, while the 0s ensure no mixing occurs between different components.

Why is this important?

The identity matrix serves several crucial purposes in mathematics:

Reference point: It provides a baseline for comparing other transformations
Matrix equations: It's essential for solving systems of equations using matrices
Inverse matrices: The identity matrix is closely related to finding matrix inverses
Verification: You can check if two matrices are inverses by seeing if their product equals the identity matrix

Important

Understanding the identity matrix is crucial because it appears in many advanced topics like finding matrix inverses, solving matrix equations, and understanding linear transformations. Master this concept early!

Common exam applications

In GCSE mathematics, you'll typically encounter the identity matrix in these contexts:

Recognising it when given in matrix form
Understanding its multiplication properties
Using it in transformation problems
Identifying when a transformation produces no change

Remember that the identity matrix is unique - there's only one 2×2 identity matrix, and it always has the same form regardless of the problem you're working on.

Summary

Key Points to Remember:

The identity matrix $\mathbf{I}$ has 1s on the main diagonal and 0s elsewhere: $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
Multiplying any matrix by the identity matrix leaves it unchanged: $\mathbf{AI} = \mathbf{IA} = \mathbf{A}$
As a transformation matrix, the identity matrix causes no movement at all
The identity matrix acts like the number 1 in regular multiplication
It's a fundamental building block for understanding more complex matrix operations

The identity matrix (AQA GCSE Further Maths): Revision Notes