Identity Matrix - Definition, Properties, Examples

What Is an Identity Matrix?

An identity matrix is a square matrix whose main-diagonal entries are all 1 and whose every other entry is 0. It is also called the unit matrix. In symbols, a_{ij} = 1 when i = j and a_{ij} = 0 when i ≠ j.

The 3x3 identity: I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}

Variable glossary. a_{ij} is the entry in row i, column j. I (or I_n for a specific size) denotes the identity matrix. The subscript n in I_n is the order, so I_2 is 2×2 and I_3 is 3×3. Because it has nonzeros only on the diagonal, the identity is a special diagonal matrix.

How Is the Identity Matrix Written?

There is one identity matrix for each size n, all denoted I or, when the size matters, I_n:

I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}, \qquad I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}

The size is fixed by context: when you multiply a 3×3 matrix by the identity, you use I_3, because the orders have to be compatible for multiplication of matrices to work.

Why Does the Identity Matrix Act Like the Number 1?

The defining property is that multiplying by I changes nothing:

AI = IA = A

For ordinary numbers, 1 is the multiplicative identity: 5×1 = 5. The matrix I plays the same role for matrices, and it is one of the few matrices that commutes with everything: AIAI and IAIA both give A, even though matrix multiplication usually depends on order.

Two more properties follow directly:

Its determinant is 1. The determinant of a diagonal matrix is the product of the diagonal entries, and 1×1×⋯×1 = 1. So ∣I∣ = 1, which confirms the identity is always non-singular.
It is its own inverse. Since II = I, the inverse of the identity is the identity: I^{-1} = I. Any power works the same way, so I^n = I.

The identity is also the destination of inversion: a matrix A times its inverse returns the identity, AA^{-1} = I. That equation is the definition of "inverse."

Examples of the Identity Matrix

Example 1

Which of these is an identity matrix? \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} or \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix}?

Check the off-diagonal entries. The first has 0s off the diagonal and 1s on it. The second has a stray 1 at position (1,2).

Final answer: Only the first is an identity matrix. The second is upper-triangular, not the identity.

Example 2

Show that AI = AA = A for A = \begin{bmatrix} 2 & 5 \ 3 & 4 \end{bmatrix} and I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}.

This is the wrong-path-first example.

Wrong attempt. A student "multiplies" by multiplying entry against entry, getting \begin{bmatrix} 2 & 0 \ 0 & 4 \end{bmatrix}, and concludes I does change A.

Why it is wrong. Matrix multiplication is row-by-column, not entry-by-entry.

Correct. Multiply each row of A against each column of I:

Top-left: (2)(1)+(5)(0)=2, Top-right: (2)(0)+(5)(1)=5, Bottom-left: (3)(1)+(4)(0)=3, Bottom-right: (3)(0)+(4)(1)=4.

AI = \begin{bmatrix} 2 & 5 \ 3 & 4 \end{bmatrix}.

Final answer: AI = AA = A, exactly as the identity property promises.

Example 3

Write the 4×4 identity matrix.

Place 1s on the main diagonal, 0s everywhere else:

I_4 = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}.

Final answer: The matrix above, I_4.

Example 4

Find the determinant of I_5.

The identity is diagonal, so its determinant is the product of the diagonal entries:

∣I_5∣=1×1×1×1×1=1.

Final answer: ∣I_5∣=1, and indeed ∣I_n∣=1 for every n.

Example 5

The matrix \begin{bmatrix} x & 0 \ 0 & y \end{bmatrix} is the 2×2 identity. Find x and y.

For the identity, the diagonal entries must both equal 1: x = 1, y = 1.

Final answer: x = 1 and y = 1.

Example 6

Confirm I^{-1} = I for I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}.

A matrix times its inverse gives the identity. Test I×I:

\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = I.

Final answer: I^{-1} = I.

Why the Identity Matrix Matters

The identity matrix exists because matrix multiplication needed a "do nothing" element to make the algebra complete. Without it, you cannot define inverses, and without inverses you cannot solve AX = B by writing X = A^{-1}B. The identity is the anchor the whole solving machinery hangs on.

Where it shows up beyond the textbook:

Solving equations. Gaussian elimination drives a matrix toward the identity; the moment the left side becomes I, the right side holds the solution.
Computer graphics. A transformation that leaves an object exactly where it is, with no rotation and no scaling, is the identity matrix.
The basis of inverses. Every invertible matrix is defined by the fact that AA^{-1} = I. The identity is the target that defines what an inverse even means.

Why Students Get the Identity Matrix Wrong

Mistake 1: Putting 1s everywhere instead of only the diagonal

Where it slips in: Writing the identity from memory.

Don't do this: Fill the whole matrix, or the first row, with 1s.

The correct way: Exactly one 1 per row, sitting on the main diagonal, and 0s everywhere else.

Mistake 2: Assuming any diagonal matrix is the identity

Where it slips in: Seeing a clean diagonal matrix and naming it I.

Don't do this: Call \begin{bmatrix} 3 & 0 \ 0 & 3 \end{bmatrix} the identity.

The correct way: The diagonal entries must all be exactly 1.

Mistake 3: Forgetting the identity must match the order for multiplication

Where it slips in: Multiplying a 3×3 matrix by I_2.

Don't do this: Use a 2×2 identity on a 3×3 matrix.

The correct way: Match the order: a 3×3 matrix needs I_3.

Key Takeaways

An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere, written I or I_n.
Multiplying any matrix by the identity leaves it unchanged: AI = IA = A
Its determinant is always 1, and it is its own inverse: I^{-1} = I.
The identity is a special diagonal and scalar matrix, with one identity matrix for each order.
Reaching the identity is the goal of inversion and elimination.