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Identity Matrices
and Basic Properties

Explore the matrix that does nothing when multiplied: the multiplicative identity of linear algebra. Learn its structure, properties, and why it matters.

Linear Algebra

What Is an Identity Matrix?

The identity matrix, written \( I_n \), is the \( n \times n \) square matrix with 1s on the main diagonal and 0s everywhere else. It is the matrix equivalent of the number 1 in ordinary multiplication.

Visualizing \( I_2 \) and \( I_3 \)

\( I_2 \)
\( 1 \)
\( 0 \)
\( 0 \)
\( 1 \)
\( I_3 \)
\( 1 \)
\( 0 \)
\( 0 \)
\( 0 \)
\( 1 \)
\( 0 \)
\( 0 \)
\( 0 \)
\( 1 \)
 Main diagonal entries are highlighted

Formal Definition

The identity matrix \( I_n \) is defined entry-by-entry using the Kronecker delta:

$$ (I_n)_{ij} = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases} $$

Why Does \( I \) Matter?

Multiplicative Identity
\( AI = IA = A \) for every conformable matrix \( A \). It is the "do nothing" operation.
Inverse Definition
A matrix \( A \) is invertible when \( A A^{-1} = I \). The identity is the target you reach.
Systems of Equations
Row-reducing an augmented matrix to \( I \) on the left side gives the solution directly.

Core Property: \( AI = IA = A \)

The fundamental property of the identity matrix is that it leaves any matrix unchanged under multiplication, provided the dimensions are compatible:

$$ A_{m \times n} \cdot I_n = A_{m \times n} \qquad I_m \cdot A_{m \times n} = A_{m \times n} $$

Multiplication in Action

\( a \)
\( b \)
\( c \)
\( d \)
\( \cdot \)
\( 1 \)
\( 0 \)
\( 0 \)
\( 1 \)
=
\( a \)
\( b \)
\( c \)
\( d \)
\( = A \)

Identity vs. Zero Matrix

Identity \( I_n \) Zero \( O_n \)
Diagonal entries 1 0
Off-diagonal entries 0 0
Role in multiplication \( AI = A \) \( AO = O \)
Real-number analogy Like multiplying by 1 Like multiplying by 0
Role in addition No special role \( A + O = A \) (additive identity)
Determinant \( \det(I) = 1 \) \( \det(O) = 0 \)

More Properties of \( I \)

1
\( I^k = I \) for any positive integer \( k \). Multiplying \( I \) by itself always gives \( I \) back.
2
\( I^{-1} = I \). The identity matrix is its own inverse.
3
\( I^T = I \). The transpose of the identity is itself (it is symmetric).
4
\( \det(I_n) = 1 \). The determinant of any identity matrix is exactly 1.
5
All eigenvalues of \( I_n \) are 1. Every nonzero vector is an eigenvector of \( I \).

How to Construct \( I_n \)

1
Determine the size \( n \). The identity matrix must be \( n \times n \) (always square).
2
Fill the main diagonal with 1s. Set entries where row index \( i \) equals column index \( j \): \( a_{11} = a_{22} = \dots = a_{nn} = 1 \).
3
Fill all other entries with 0. Every off-diagonal entry \( a_{ij} \) with \( i \neq j \) is set to 0.
4
Verify: Check that multiplying your matrix by itself gives the same matrix back: \( I \cdot I = I \).

Example 1: \( 2 \times 2 \) Multiplication

Compute \( A \cdot I_2 \) where $$ A = \begin{pmatrix} 3 & 5 \\ -1 & 4 \end{pmatrix} $$.

\( 3 \)
\( 5 \)
\( -1 \)
\( 4 \)
\( \cdot \)
\( 1 \)
\( 0 \)
\( 0 \)
\( 1 \)
=
\( 3 \)
\( 5 \)
\( -1 \)
\( 4 \)

Example 2: \( 3 \times 3 \) Identity

Verify that \( I_3 \cdot B = B \) where $$ B = \begin{pmatrix} 2 & 0 & -1 \\ 7 & 3 & 5 \\ -4 & 1 & 6 \end{pmatrix} $$.

1
Row 1 of \( I_3 \): \( (1, 0, 0) \). Dot with each column of \( B \) selects row 1 of \( B \): \( (2, 0, -1) \).
2
Row 2 of \( I_3 \): \( (0, 1, 0) \). Selects row 2 of \( B \): \( (7, 3, 5) \).
3
Row 3 of \( I_3 \): \( (0, 0, 1) \). Selects row 3 of \( B \): \( (-4, 1, 6) \).

Example 3: Non-square Matrices

Let $$ C = \begin{pmatrix} 1 & 4 & 2 \\ 3 & -2 & 5 \end{pmatrix} $$, a \( 2 \times 3 \) matrix. Which identity works?

Right multiply
\( C \cdot I_3 = C \). Use \( I_3 \) because \( C \) has 3 columns.
Left multiply
\( I_2 \cdot C = C \). Use \( I_2 \) because \( C \) has 2 rows.

Identity and Inverses

A square matrix \( A \) is invertible if there exists a matrix \( A^{-1} \) such that:

$$ A A^{-1} = A^{-1} A = I_n $$
Existence
\( A^{-1} \) exists if and only if \( \det(A) \neq 0 \).
Uniqueness
If \( A^{-1} \) exists, it is unique: there is exactly one inverse.

Common Mistakes

✕ Using the wrong size
Trying \( A_{2 \times 3} \cdot I_2 \) instead of \( A_{2 \times 3} \cdot I_3 \). The inner dimensions must match.
✕ Non-square "identity"
Writing a \( 2 \times 3 \) matrix with 1s on the diagonal. The identity must always be square.
✕ Confusing \( I \) and \( O \)
Thinking \( AI = O \) (zero matrix). The identity preserves \( A \); the zero matrix annihilates it.

Summary of Properties

Property Statement
Multiplicative identity \( AI = IA = A \)
Self-inverse \( I^{-1} = I \)
Symmetric \( I^T = I \)
Idempotent \( I^k = I \) for all \( k \geq 1 \)
Determinant \( \det(I_n) = 1 \)
Eigenvalues All eigenvalues equal 1
Trace \( \text{tr}(I_n) = n \)

Question 1 of 5

What is the entry in row 2, column 3 of \( I_4 \)?

Question 2 of 5

If \( A \) is a \( 3 \times 5 \) matrix, which identity matrix satisfies \( A \cdot I = A \)?

Question 3 of 5

What is \( \det(I_7) \)?

Question 4 of 5

Which statement about the identity matrix is false?

Question 5 of 5

If \( A \) is invertible and \( AB = I \), what is \( B \)?