Unit Matrix — Definition, Formula & Examples

A unit matrix is a square matrix that has 1 in every diagonal position and 0 in every off-diagonal position. It is the standard synonym for the identity matrix, often denoted

I

I_n

The unit matrix of order $n$ , denoted $I_n$ , is the $n \times n$ matrix whose entries satisfy $\delta_{ij}$ , the Kronecker delta: the entry in row $i$ , column $j$ equals 1 when $i = j$ and 0 when $i \neq j$ . It serves as the multiplicative identity in the ring of $n \times n$ matrices.

Key Formula

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

Where:

$I_n$ = The unit (identity) matrix of size n × n
$i, j$ = Row and column indices, respectively
$\delta_{ij}$ = Kronecker delta, equal to 1 when i = j and 0 otherwise

Worked Example

Problem: Verify that multiplying the 3×3 unit matrix by a matrix A returns A, where A = [[2, 5, 1], [0, 3, 4], [7, 6, 8]].

Write the 3×3 unit matrix: Place 1s along the main diagonal and 0s everywhere else.

I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

Multiply I₃ · A: Each row of I₃ picks out exactly the corresponding row of A, leaving every entry unchanged.

I_3 \cdot A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} 2 & 5 & 1 \\ 0 & 3 & 4 \\ 7 & 6 & 8 \end{pmatrix} = \begin{pmatrix} 2 & 5 & 1 \\ 0 & 3 & 4 \\ 7 & 6 & 8 \end{pmatrix}

Answer:

I_3 \cdot A = A

, confirming that the unit matrix acts as the multiplicative identity.

Why It Matters

The unit matrix appears whenever you solve systems via row reduction — you reduce the coefficient matrix to

I_n

to read off solutions. It is also central to computing matrix inverses: if

AB = I_n

, then

B = A^{-1}

. In applied fields like computer graphics and machine learning, identity matrices initialize transformations and weight matrices.

Common Mistakes

Mistake: Confusing the unit matrix with a matrix of all 1s.

Correction: A unit matrix has 1s only on the main diagonal. The off-diagonal entries are all 0. A matrix filled entirely with 1s is sometimes called a ones matrix or matrix of ones — it is not the identity.