Basic

Identity Matrix Property

Multiplying by the identity leaves a matrix unchanged.

Formula

AI = IA = A

Variables

AMatrix

IIdentity

Example

Like multiplying by 1

How it works

The identity matrix is the matrix version of "do nothing." Its 1s on the main diagonal and 0s everywhere else mean that, when it multiplies a matrix, each entry is copied into the same position without being mixed with other entries. This works because matrix multiplication forms each output entry from dot products, and the identity matrix makes those dot products pick out exactly one matching entry. As a result, both left-multiplying and right-multiplying by the identity leave the matrix unchanged.

When to use it

Use this property whenever you need to simplify matrix expressions, check algebraic steps, or verify that a matrix inverse is correct. It is also useful in linear systems, computer graphics, and any situation where you want to confirm that a transformation leaves vectors unchanged.

Worked example, step by step

Let A = [[2, -1], [5, 3]] and let I = [[1, 0], [0, 1]]. Compute IA: IA = [[1, 0], [0, 1]] [[2, -1], [5, 3]] = [[1·2 + 0·5, 1·(-1) + 0·3], [0·2 + 1·5, 0·(-1) + 1·3]] = [[2, -1], [5, 3]] Compute AI: AI = [[2, -1], [5, 3]] [[1, 0], [0, 1]] = [[2·1 + (-1)·0, 2·0 + (-1)·1], [5·1 + 3·0, 5·0 + 3·1]] = [[2, -1], [5, 3]] So IA = AI = A.

Common mistakes to avoid

Using a non-square matrix as the identity matrix
Thinking the identity only works on the left or only on the right
Forgetting that the identity matrix must match the size of A

Did You Know?

The identity matrix is the number 1 of matrix multiplication: anything times it stays unchanged.

matrix

Identity Matrix Property

Formula

Variables

Example

How it works

When to use it

Worked example, step by step

Common mistakes to avoid

Did You Know?

Share this formula

More in Linear Algebra

Determinant (2×2)

Dot Product

Cross Product Magnitude

Vector Magnitude

Eigenvalue Equation

Inverse of a 2×2 Matrix