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Linear Algebra • Systems of Linear Equations & Matrices

Identity Matrices

The special matrix that leaves every other matrix unchanged — the "1" of the matrix world.

Explore the lesson 14 min read
Linear Algebra emblem with a matrix grid, a variable, and three-dimensional axes.
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Historical Context & Motivation

Imagine you're multiplying a number by 1. The answer is always the same number you started with. Mathematicians wanted to find something that works the same way for matrices (rectangular grids of numbers). That search led to one of the most important building blocks in all of linear algebra: the identity matrix.

The story of matrices stretches back centuries, growing out of efforts to solve systems of equations and to organize data. As mathematicians developed rules for adding and multiplying matrices, they realized they needed a special matrix that acted like the number 1 does for regular multiplication. Let's see how that idea developed over time.

1683 • Seki Takakazu & Determinants

Japanese mathematician Seki Takakazu introduced early ideas about arrays of numbers (determinants), laying groundwork for matrix theory long before the word "matrix" existed.

1850 • James Joseph Sylvester Coins "Matrix"

English mathematician Sylvester used the word "matrix" to describe a rectangular arrangement of numbers, giving the concept its modern name.

1858 • Arthur Cayley Defines Matrix Algebra

Cayley published the first systematic rules for matrix multiplication and explicitly described the identity matrix as the element that leaves any matrix unchanged when multiplied.

1900s • Matrices Enter Physics & Engineering

Identity matrices became essential in quantum mechanics, computer graphics, and data science — anywhere transformations need a "do nothing" starting point.

So here's the question this lesson answers: What does a matrix look like if multiplying it by any other matrix gives back the original matrix? That special matrix is the identity matrix, and understanding it is the key to unlocking inverse matrices, solving systems of equations, and much more.

SECTION 2

Core Principles & Definitions

An identity matrix is a square matrix (same number of rows and columns) that has 1s along its main diagonal and 0s everywhere else. It is usually written as I, sometimes with a subscript showing its size (like I₃ for a 3 × 3 identity matrix). Here are the foundational ideas you need to know.

1

Square Shape

An identity matrix must be square — it has the same number of rows as columns. Common sizes are 2 × 2, 3 × 3, and 4 × 4.
2

Ones on the Diagonal

The main diagonal runs from the top-left corner to the bottom-right corner. Every entry on this diagonal is 1.
3

Zeros Everywhere Else

Every entry that is NOT on the main diagonal is 0. These zeros are what make the identity matrix special.
4

Multiplicative Identity Property

When you multiply any matrix A by the identity matrix I, the result is just A. In symbols: A × I = A and I × A = A.
5

Unique for Each Size

There is exactly one identity matrix for every size. The 2 × 2 identity matrix is different from the 3 × 3 identity matrix, but each one plays the same role for matrices of its size.
✦

KEY TAKEAWAY

Think of the identity matrix like a mirror at a photo booth that doesn't add any filters or effects. When a matrix "looks into" the identity matrix through multiplication, it sees an exact copy of itself — nothing changes. Just as multiplying any number by 1 gives you the same number, multiplying any matrix by the identity matrix gives you the same matrix.

SECTION 3

Visual Explanation

The diagram below shows what 2 × 2, 3 × 3, and 4 × 4 identity matrices look like. Notice the clear pattern: 1s march along the main diagonal from top-left to bottom-right, and every other entry is 0. No matter how large the identity matrix gets, this pattern holds.

Identity Matrices: I₂ , I₃ , and I₄I₂ (2 × 2)[][]1001I₃ (3 × 3)100010001I₄ (4 × 4)1000010000100001Dashed line = main diagonal Highlighted cells = diagonal entries (all 1s) All other entries = 0
Each identity matrix has 1s on the main diagonal (shown by the highlighted cells and dashed line) and 0s everywhere else. The pattern is the same no matter how large the matrix gets.

In the diagram above, the main diagonal is the line of entries running from the top-left corner to the bottom-right corner. Every entry on this diagonal is 1, and every entry off the diagonal is 0. You can write any identity matrix by following this simple rule, whether the matrix is 2 × 2 or 100 × 100.

SECTION 4

Mathematical Framework

Now let's express the identity matrix and its key property using mathematical notation. Don't worry — we'll break every part down step by step.

IDENTITY MATRIX DEFINITION

In=[δij]whereδij=1ifi=j,δij=0ifi≠jIₙ = [δᵢⱼ] where δᵢⱼ = 1 if i = j, δᵢⱼ = 0 if i ≠ j

Here, n is the size of the matrix, i is the row number, and j is the column number. The symbol δᵢⱼ (called the Kronecker delta) simply says: "put a 1 if the row and column numbers match, and a 0 if they don't."

MULTIPLICATIVE IDENTITY PROPERTY

A×In=AandIn×A=AA × Iₙ = A and Iₙ × A = A

For any matrix A whose dimensions are compatible, multiplying by Iₙ returns A unchanged. This is why it's called the "identity" — it preserves the identity of any matrix it touches.

2 × 2 IDENTITY MATRIX

I2=[10][01]I₂ = [ 1 0 ] [ 0 1 ]

This is the smallest identity matrix. It has 1s at positions (1,1) and (2,2) and 0s at (1,2) and (2,1).

RELATIONSHIP TO INVERSE MATRICES

A×A−1=IandA−1×A=IA × A⁻¹ = I and A⁻¹ × A = I

The identity matrix appears whenever you talk about inverse matrices. If a matrix A has an inverse A⁻¹, then multiplying them together gives the identity matrix. Think of it like: 5 × (1/5) = 1.

The key idea to remember is that the identity matrix plays the exact same role in matrix multiplication that the number 1 plays in regular multiplication. Whenever you see I in a matrix equation, you can think of it as the "do nothing" matrix.

SECTION 5

How Matrix Multiplication by I Works

Let's see why multiplying a matrix by the identity matrix gives back the original. The diagram below walks through a 2 × 2 example, showing each dot product that makes up the result.

Multiplying Matrix A by I₂ Step by StepEach entry of the result = dot product of a row of A with a column of I₂Matrix A3725×I₂1001=Result = A3725Entry (1,1):Row 1 of A · Col 1 of I = (3 × 1) + (7 × 0) = 3 + 0 = 3 ✓Entry (1,2):Row 1 of A · Col 2 of I = (3 × 0) + (7 × 1) = 0 + 7 = 7 ✓Entry (2,1):Row 2 of A · Col 1 of I = (2 × 1) + (5 × 0) = 2 + 0 = 2 ✓Entry (2,2):Row 2 of A · Col 2 of I = (2 × 0) + (5 × 1) = 0 + 5 = 5 ✓
When you multiply matrix A = [3, 7; 2, 5] by I₂, each dot product picks out exactly one entry from A because the identity column has only one 1 in it. The result is the original matrix A.

Here's the magic: when you take the dot product of any row of A with a column of I, the 1 in the identity column "selects" exactly one element from that row, and the 0s cause the other elements to vanish. That's why the result is always the original matrix. The identity matrix acts like a filter that lets everything through unchanged.

💡

PATTERN INSIGHT

Column j of the identity matrix has a single 1 in row j. When you dot any row with that column, the 1 picks out the entry in position j and the 0s eliminate everything else. This is why every entry of the product lands back in its original spot.

SECTION 6

Worked Example

Let's work through a full example with a 3 × 3 matrix to make sure the identity property holds for larger matrices too.

Verify that B × I₃ = B

  1. 1

    Step 1 — Write Down Matrix B and I₃

    Let B = [ 4 −1 6 ; 0 3 2 ; 5 −2 8 ] and I₃ = [ 1 0 0 ; 0 1 0 ; 0 0 1 ]. We will multiply B × I₃ and check that the result equals B.
  2. 2

    Step 2 — Compute Row 1 of the Product

    Row 1 of B is [4, −1, 6]. We dot it with each column of I₃. Column 1: (4 × 1) + (−1 × 0) + (6 × 0) = 4. Column 2: (4 × 0) + (−1 × 1) + (6 × 0) = −1. Column 3: (4 × 0) + (−1 × 0) + (6 × 1) = 6.
    Row 1 of the product = [4, −1, 6] ✓ matches Row 1 of B.
  3. 3

    Step 3 — Compute Row 2 of the Product

    Row 2 of B is [0, 3, 2]. Column 1: (0 × 1) + (3 × 0) + (2 × 0) = 0. Column 2: (0 × 0) + (3 × 1) + (2 × 0) = 3. Column 3: (0 × 0) + (3 × 0) + (2 × 1) = 2.
    Row 2 of the product = [0, 3, 2] ✓ matches Row 2 of B.
  4. 4

    Step 4 — Compute Row 3 of the Product

    Row 3 of B is [5, −2, 8]. Column 1: (5 × 1) + (−2 × 0) + (8 × 0) = 5. Column 2: (5 × 0) + (−2 × 1) + (8 × 0) = −2. Column 3: (5 × 0) + (−2 × 0) + (8 × 1) = 8.
    Row 3 of the product = [5, −2, 8] ✓ matches Row 3 of B.
  5. 5

    Step 5 — State the Conclusion

    Combining all three rows, the product B × I₃ = [ 4 −1 6 ; 0 3 2 ; 5 −2 8 ] = B. The identity matrix preserved every entry of B, exactly as predicted by the identity property.
    B × I₃ = B ✓ Verified!

SECTION 7

Identity Matrix vs. Other Special Matrices

The identity matrix is one of several special matrices you'll encounter in linear algebra. The table below compares it to other common types so you can see how they're alike and how they differ.

Matrix TypeWhat Makes It SpecialRelationship to I
Identity Matrix (I)1s on the main diagonal, 0s everywhere elseIt IS the identity! A × I = A.
Zero Matrix (O)Every entry is 0A × O = O. The zero matrix annihilates — the opposite of I.
Diagonal MatrixNon-zero entries only on the diagonal (values can differ)I is a special diagonal matrix where every diagonal entry equals 1.
Scalar MatrixSame constant on the diagonal, 0s elsewhere (like 3I)A scalar matrix is just a constant times I. For example, 3I₂ = [3,0;0,3].
Inverse Matrix (A⁻¹)"Undoes" A so that A × A⁻¹ = IThe product of a matrix and its inverse is always I.

Comparison of the identity matrix with other special matrices

✦

KEY TAKEAWAY

The identity matrix is to matrix multiplication what the number 1 is to regular multiplication, and what the number 0 is to addition. Each of these is a neutral element — it lets you perform the operation without changing the value you started with. Recognizing neutral elements is a recurring theme in algebra and will come up again when you study groups and rings in advanced math.

SECTION 8

Connection to Advanced Topics

The identity matrix isn't just a classroom concept — it shows up everywhere in more advanced math and real-world applications. Here's a quick look at where the identity matrix leads.

Concept in This LessonAdvanced ExtensionWhy It Matters
A × I = AInverse matrices: A × A⁻¹ = ISolving systems of equations by finding a matrix that "undoes" another
Diagonal of 1sDeterminants: det(I) = 1The determinant of the identity matrix is always 1, a benchmark for understanding other determinants
I preserves a matrixEigenvalues: det(A − λI) = 0I appears in the characteristic equation used to find eigenvalues, which reveal stretching behavior of transformations
Neutral element for multiplicationComputer graphicsTransformations start from the identity matrix; rotations, scaling, and translations are built by modifying I

How identity matrices connect to advanced topics

As you move deeper into linear algebra, you'll find the identity matrix hiding inside almost every major formula. Mastering it now gives you a strong foundation for topics like inverse matrices, eigenvalues, and matrix transformations in computer graphics and engineering.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In your own words, explain why the identity matrix is called the "identity" matrix. What does it do when multiplied by another matrix, and what everyday number does it behave like?
Show answer
The identity matrix is called the "identity" matrix because it preserves the identity of any matrix it multiplies. When you compute A × I or I × A, the result is simply A — every entry stays exactly where it was. This is just like multiplying any number by 1: for example, 7 × 1 = 7. The number 1 doesn't change anything in regular multiplication, and the identity matrix doesn't change anything in matrix multiplication. In both cases, the element acts as a neutral or "do nothing" operation.

Question 2: BASIC CALCULATION

Write out the 3 × 3 identity matrix I₃. Then compute the product I₃ × C, where C = [ 2 0 −4 ; 1 5 3 ; −3 7 6 ].
Show answer
The 3 × 3 identity matrix is I₃ = [ 1 0 0 ; 0 1 0 ; 0 0 1 ]. By the multiplicative identity property, I₃ × C = C. You can verify this by performing each dot product: Row 1 of I₃ is [1, 0, 0], and dotting it with each column of C picks out the first entry of each column, giving [2, 0, −4]. Row 2 of I₃ is [0, 1, 0], which picks out the second entry of each column, giving [1, 5, 3]. Row 3 of I₃ is [0, 0, 1], which picks out the third entry, giving [−3, 7, 6]. So the result is [ 2 0 −4 ; 1 5 3 ; −3 7 6 ] = C.

Question 3: INTERMEDIATE

A student claims that the matrix M = [ 1 0 ; 0 0 ] is a 2 × 2 identity matrix. Is this correct? Explain your reasoning by multiplying M by A = [ 3 5 ; 2 4 ] and checking whether M × A = A.
Show answer
The student is incorrect. The matrix M = [ 1 0 ; 0 0 ] is NOT the 2 × 2 identity matrix because the entry in position (2,2) is 0 instead of 1. To confirm, compute M × A: Row 1 of M is [1, 0], so (1 × 3) + (0 × 2) = 3 and (1 × 5) + (0 × 4) = 5, giving Row 1 = [3, 5]. Row 2 of M is [0, 0], so (0 × 3) + (0 × 2) = 0 and (0 × 5) + (0 × 4) = 0, giving Row 2 = [0, 0]. The product M × A = [ 3 5 ; 0 0 ], which does NOT equal A = [ 3 5 ; 2 4 ]. The second row was wiped out because M had a 0 instead of a 1 on the diagonal. The correct I₂ = [ 1 0 ; 0 1 ].

Question 4: APPLIED

In a video game, a character's position is stored as a 2 × 1 column vector. Multiplying this vector by a 2 × 2 transformation matrix moves the character. If the game needs a "no movement" option, which 2 × 2 matrix should it use? Write out this matrix and explain why it guarantees the character stays in place.
Show answer
The game should use the 2 × 2 identity matrix I₂ = [ 1 0 ; 0 1 ]. When you multiply I₂ by any position vector [x ; y], you get: (1 × x) + (0 × y) = x for the first entry, and (0 × x) + (1 × y) = y for the second entry. The result is [x ; y], the original position — the character hasn't moved at all. This is why the identity matrix is the default "starting" transformation in computer graphics: it represents doing nothing to the object. Every other transformation (rotation, scaling, reflection) is defined as a modification of the identity matrix.

Question 5: CRITICAL THINKING

Suppose you have a matrix A and you know that A × B = I and B × A = I. What can you conclude about the relationship between A and B? Could there be more than one matrix B with this property for a given A? Explain your reasoning.
Show answer
If A × B = I and B × A = I, then B is the inverse of A (written B = A⁻¹). This means B "undoes" whatever A does, and vice versa. A key theorem in linear algebra states that if a square matrix has an inverse, that inverse is unique — there is exactly one matrix B satisfying both equations. Here is a quick sketch of why: suppose there were two inverses, B and C, so that A × B = I and C × A = I. Then C = C × I = C × (A × B) = (C × A) × B = I × B = B, proving C = B. So no, there cannot be more than one matrix B with this property for a given A. This uniqueness is one of the most important results in matrix algebra and depends directly on the properties of the identity matrix.

SUMMARY

Lesson Summary

Summary

The identity matrix (written as I or Iₙ) is a square matrix with 1s on the main diagonal and 0s everywhere else. Its defining property is the multiplicative identity property: multiplying any compatible matrix A by I gives back A (A × I = A and I × A = A). It plays the same role for matrices that the number 1 plays for ordinary numbers.

The identity matrix is the foundation for understanding inverse matrices (A × A⁻¹ = I), determinants (det(I) = 1), and matrix transformations in fields like computer graphics and physics. Whenever you see I in an equation, remember: it means "leave everything unchanged."

Varsity Tutors • Linear Algebra • Identity Matrices