What is the identity matrix?

The identity matrix, written I, is a square matrix with 1s on the main diagonal and 0s everywhere else. For any matrix A, multiplying by I does not change A: we always get IA = AI = A. Think of it as the 'do nothing' transformation in matrix algebra.

Why is it called the identity matrix?

It is called 'identity' because its only job is to preserve the identity—the original value—of whatever matrix it multiplies. Just as multiplying a number by 1 does not change it, multiplying a matrix by the identity matrix does not change it either.

Can you multiply any matrix by the identity?

Multiplying A by I only works if the dimensions match: if A is m \times n, you multiply by an n \times n identity on the right to get AI, or an m \times m identity on the left to get IA. The key rule is that the inner dimensions must match for matrix multiplication to be defined.

What is the identity matrix used for in real life?

The identity matrix is essential for finding matrix inverses and solving equations like AX = B. It is also used to test whether matrix calculations are correct, for computer graphics where you might start with no transformation and then apply rotations, and whenever you need to represent 'do nothing' as a transformation.

Identity Matrix

The identity matrix leaves any matrix unchanged when multiplied—the matrix equivalent of 1, essential for solving linear systems and finding inverses.

AI = IA = A

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What each symbol means

What Identity Matrix takes

Symbol	Meaning
$A$	A is any matrix being multiplied. The identity matrix leaves $A$ completely unchanged, so $AI = A$ and $IA = A$ . Misreading $A$ as a number instead of a matrix leads students to forget they are doing matrix multiplication, not scalar multiplication.
$I$	I is the identity matrix, a square matrix with 1s on the main diagonal (top-left to bottom-right) and 0s in all other positions. If you misread $I$ as the number 1, you will not understand why matrix multiplication by $I$ works so differently from regular multiplication by 1.

When to use it

Reach for the identity matrix when you are solving matrix equations, finding matrix inverses, or need to verify that your matrix calculations are correct.

Level

Usually taught in: Algebra II

Worked examples

1. Multiply a matrix by the 2×2 identity matrix

Problem

Multiply

A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}

by the identity matrix

I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

and show that

AI = A

$\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
Set up the multiplication: each entry in the result comes from multiplying a row of $A$ by a column of $I$ .
$3 \cdot 1 + 2 \cdot 0 = 3$
Multiply row 1 of $A$ by column 1 of $I$ : the first entry is $3 \cdot 1 = 3$ .
$3 \cdot 0 + 2 \cdot 1 = 2$
Multiply row 1 of $A$ by column 2 of $I$ : the second entry is $2 \cdot 1 = 2$ .
$1 \cdot 1 + 4 \cdot 0 = 1$
Multiply row 2 of $A$ by column 1 of $I$ : the third entry is $1 \cdot 1 = 1$ .
$1 \cdot 0 + 4 \cdot 1 = 4$
Multiply row 2 of $A$ by column 2 of $I$ : the fourth entry is $4 \cdot 1 = 4$ .

Answer: $\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$

The result is exactly $A$ . This is what $AI = A$ means: multiplying by the identity leaves a matrix unchanged, just as multiplying a number by 1 leaves it unchanged.

2. Verify $IA = A$ with negative and fractional entries

Problem

Show that

I \begin{pmatrix} -2 & \frac{1}{2} \\ 3 & -4 \end{pmatrix} = \begin{pmatrix} -2 & \frac{1}{2} \\ 3 & -4 \end{pmatrix}

where

I

is the

2 \times 2

identity matrix.

$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} -2 & \frac{1}{2} \\ 3 & -4 \end{pmatrix}$
Set up the multiplication with $I$ on the left and $A$ on the right.
$1 \cdot (-2) + 0 \cdot 3 = -2 \quad \text{and} \quad 1 \cdot \frac{1}{2} + 0 \cdot (-4) = \frac{1}{2}$
The first row of $I$ is $[1, 0]$ , so it picks out the first row of $A$ : $-2$ and $\frac{1}{2}$ .
$0 \cdot (-2) + 1 \cdot 3 = 3 \quad \text{and} \quad 0 \cdot \frac{1}{2} + 1 \cdot (-4) = -4$
The second row of $I$ is $[0, 1]$ , so it picks out the second row of $A$ : $3$ and $-4$ .

Answer: $\begin{pmatrix} -2 & \frac{1}{2} \\ 3 & -4 \end{pmatrix}$

$IA = A$ , confirming that the identity works whether it is on the left or the right of $A$ . Even with negative and fractional entries, multiplying by $I$ leaves $A$ completely unchanged.

3. Track sales at three coffee shops using matrix multiplication by the identity

Problem

A coffee shop chain has three locations. Monday's sales were recorded in the matrix

S = \begin{pmatrix} 340 & 85 \\ 290 & 110 \\ 320 & 95 \end{pmatrix}

, where rows are locations and columns are coffee (in dollars) and pastries (in dollars). Multiply

S

by the

2 \times 2

identity matrix to verify that

SI = S

$\begin{pmatrix} 340 & 85 \\ 290 & 110 \\ 320 & 95 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
Set up the multiplication: $S$ is $3 \times 2$ and $I$ is $2 \times 2$ , so the result will be $3 \times 2$ .
$340 \cdot 1 + 85 \cdot 0 = 340 \quad \text{and} \quad 340 \cdot 0 + 85 \cdot 1 = 85$
Location 1's sales of $340 for coffee and $85 for pastries remain exactly the same after multiplying by $I$ .
$290 \cdot 1 + 110 \cdot 0 = 290 \quad \text{and} \quad 290 \cdot 0 + 110 \cdot 1 = 110$
Location 2's sales of $290 and $110 stay unchanged.
$320 \cdot 1 + 95 \cdot 0 = 320 \quad \text{and} \quad 320 \cdot 0 + 95 \cdot 1 = 95$
Location 3's sales of $320 and $95 remain the same.

Answer: $\begin{pmatrix} 340 & 85 \\ 290 & 110 \\ 320 & 95 \end{pmatrix}$

The result $SI$ equals $S$ : all three locations' sales data is unchanged. Multiplying by the identity is like saying 'apply no transformation'—in business applications, this ensures that data stays exactly what it was when you are organizing figures or testing whether calculations are correct.

Common mistakes

Where Identity Matrix usually goes wrong

The mistake	Why it is wrong	The fix
Writing $AI = A$ and claiming the answer without showing the row-by-column multiplication work.	If you do not carry out each entry of the multiplication step by step, you might write the correct shape of matrix but with wrong values—or you might skip multiplying entirely and miss the point that $I$ actually preserves $A$ .	Always multiply row-by-row and column-by-column, writing out at least a few entries like $(a_1)(1) + (a_2)(0) = a_1$ to show that each entry of the result matches the original.
Claiming that $IA \ne AI$ because 'matrix multiplication is not commutative', so the identity works only on one side.	The identity matrix is the one exception to the non-commutative rule: $IA = AI = A$ always holds. Many students remember that matrix multiplication is not commutative in general and incorrectly assume this means the identity also depends on order.	Verify with a concrete example: multiply a small matrix by $I$ on both the left and the right, and you will see both products equal the original matrix. This is a defining property of the identity.
Confusing the identity matrix $I$ with the scalar 1 and writing $A \cdot 1 = 1$ instead of $AI = A$ .	$I$ is a matrix, not a number. When you write $AI$ , you are doing matrix multiplication (row times column), not scalar multiplication or trying to simplify away the identity.	Remember that $I$ is always a square matrix with 1s on the main diagonal and 0s everywhere else. Matrix multiplication $AI$ follows the row-by-column rule, never element-by-element scaling.

Tips and when to use something else

The identity matrix is different for different sizes: the $2 \times 2$ identity has four entries, the $3 \times 3$ has nine, and so on. Always make sure your identity matrix matches the dimensions needed for the multiplication.
Multiplying by the identity is useful for solving matrix equations like $AX = B$ . If you find a matrix $A^{-1}$ (called the inverse) such that $A^{-1}A = I$ , you can solve for $X = A^{-1}B$ . Study matrix inverses when you are ready to solve systems this way.
The identity matrix is square: it always has the same number of rows and columns. If you have a non-square matrix like $3 \times 2$ , you will need different identity matrices on the left and right depending on which product you want.
If you want to combine transformations in geometry or graphics (rotations, scalings, reflections), you will use different matrices instead of the identity. Look into rotation matrices and scaling matrices once you are comfortable with the identity.

Related formulas

Matrix MultiplicationMatrix multiplication combines rows from the first matrix with columns of the second to produce a new matrix, useful for solving systems and transformations.Inverse of a 2x2 MatrixCompute the unique 2×2 matrix that reverses multiplication by another 2×2 matrix, essential for solving systems of linear equations.Gaussian EliminationGaussian Elimination transforms a system of linear equations into row echelon form to solve for unknown variables systematically.Determinant of a 2x2 MatrixA single number computed from a 2x2 matrix that tells whether the matrix is invertible and helps solve systems of linear equations.

Frequently asked questions

What is the identity matrix?: The identity matrix, written $I$ , is a square matrix with 1s on the main diagonal and 0s everywhere else. For any matrix $A$ , multiplying by $I$ does not change $A$ : we always get $IA = AI = A$ . Think of it as the 'do nothing' transformation in matrix algebra.
Why is it called the identity matrix?: It is called 'identity' because its only job is to preserve the identity—the original value—of whatever matrix it multiplies. Just as multiplying a number by 1 does not change it, multiplying a matrix by the identity matrix does not change it either.
Can you multiply any matrix by the identity?: Multiplying $A$ by $I$ only works if the dimensions match: if $A$ is $m \times n$ , you multiply by an $n \times n$ identity on the right to get $AI$ , or an $m \times m$ identity on the left to get $IA$ . The key rule is that the inner dimensions must match for matrix multiplication to be defined.
What is the identity matrix used for in real life?: The identity matrix is essential for finding matrix inverses and solving equations like $AX = B$ . It is also used to test whether matrix calculations are correct, for computer graphics where you might start with no transformation and then apply rotations, and whenever you need to represent 'do nothing' as a transformation.

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Reviewed 2026-09-18