Identity Matrix Property Equation, Derivation & Rearrangements

Core idea

Overview

The identity matrix, denoted as I, serves as the multiplicative identity element in linear algebra, analogous to the number 1 in scalar arithmetic. Because matrix multiplication is generally not commutative, the identity property explicitly confirms that multiplication by I commutes with A, leaving the original transformation unchanged. It acts as the 'do-nothing' operator in matrix space.

When to use: Use this property to simplify complex matrix algebraic expressions or when verifying if a matrix is an inverse of another.

Why it matters: It is fundamental for solving matrix equations of the form AX = B and serves as the basis for defining the inverse matrix A^-1, where AA^-1 = I.

Symbols

Variables

A = Matrix A, I = Identity Matrix, \mathbf{A}\mathbf{I} = \mathbf{A}\mathbf{I}

A

Matrix A

V a r iab l e

I

Identity Matrix

V a r iab l e

AI

\mathbf{A}\mathbf{I}

V a r iab l e

Walkthrough

Derivation

Derivation of Identity Matrix Property

This derivation demonstrates that the identity matrix acts as the multiplicative identity in matrix algebra by showing how its unique structure preserves the elements of a matrix during multiplication.

Matrix A is an m x n matrix with elements $a_{ij}$ .
The identity matrix $I_{n}$ is an n x n square matrix with ones on the leading diagonal and zeros elsewhere.

1

Defining the Identity Matrix

We define the identity matrix using the Kronecker delta, where the element at row i and column j is 1 only when i equals j, and 0 otherwise.

I_{ij} = δ_{ij} = {1 if i = j 0 if i \neq = j

Note: The identity matrix must always be square, whereas A can be any shape as long as dimensions are conformable.

2

Applying Matrix Multiplication

Using the definition of matrix multiplication, the element at (i,j) of the product AI is the dot product of the i-th row of A and the j-th column of I.

(A I)_{ij} = k = 1 \sum n a_{ik} I_{k j}

Note: Ensure the number of columns in A equals the number of rows in I.

3

Simplifying via Kronecker Delta

Since the summation collapses to only the term where k = j, the product simplifies to $a_{ij}$ , effectively recovering the original matrix element.

(A I)_{ij} = k = 1 \sum n a_{ik} δ_{k j} = a_{ij}

Note: This identity holds for both pre-multiplication (IA) and post-multiplication (AI).

Result

(A I)_{ij} = k = 1 \sum n a_{ik} δ_{k j} = a_{ij}

Source: Edexcel A-Level Mathematics: Pure Mathematics Year 2, Chapter 7 (Matrices)

Why it behaves this way

Intuition

Think of the identity matrix as a 'do-nothing' transformation in geometry. If matrix A represents a transformation (like stretching, rotating, or shearing a shape), multiplying by the identity matrix I is like applying a transformation that maps every point exactly to its current location—like standing in front of a mirror that reflects the world exactly as it is without shifting anything.

A

Square Matrix

An operator that reconfigures the space or coordinates of a vector.

I

Identity Matrix

The 'unit' element of matrices, containing 1s on the main diagonal and 0s elsewhere; it represents the absence of change.

Signs and relationships

=: Denotes equivalence in outcome; the transformation A remains unchanged after the operation.
AI = IA: Demonstrates commutativity specifically with the identity; while matrix multiplication is generally not commutative, the identity matrix serves as the neutral element that commutes with all conformable matrices.

One free problem

Practice Problem

If matrix A = [[2, 3], [4, 5]], what is the result of multiplying A by the 2x2 identity matrix I?

a112

a123

a214

a225

Solve for: $A$

Hint: The identity matrix property states AI = A.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In computer graphics, applying an identity matrix transformation to a 3D model results in no change to the object's position, orientation, or scale.

Study smarter

Tips

Ensure the dimensions are conformable (the number of columns in A must match the size of I).
Remember that the identity matrix must be a square matrix.
The identity matrix always has 1s on the main diagonal and 0s elsewhere.

Avoid these traps

Common Mistakes

Assuming the identity matrix has the same dimensions as A if A is not square.
Forgetting that I must be square even if A is rectangular.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation demonstrates that the identity matrix acts as the multiplicative identity in matrix algebra by showing how its unique structure preserves the elements of a matrix during multiplication.

Use this property to simplify complex matrix algebraic expressions or when verifying if a matrix is an inverse of another.

It is fundamental for solving matrix equations of the form AX = B and serves as the basis for defining the inverse matrix A^-1, where AA^-1 = I.

Assuming the identity matrix has the same dimensions as A if A is not square. Forgetting that I must be square even if A is rectangular.