Identity Matrix: Definition, Properties, and Examples

An identity matrix is the matrix counterpart of the number one in multiplication. It is square, has ones on its main diagonal and zeros everywhere else. Multiplying by a compatible identity matrix leaves the other matrix unchanged. Its order matters: a rectangular matrix may need an identity of one order on the left and an identity of another order on the right.

The subscripts describe the required dimensions

If $A$ has $m$ rows and $n$ columns, then $I_{m}A=A$ and $A{I}_n=A$. The left identity must have $m$ rows and columns so its columns match the rows of $A$. The right identity must have order $n$. Writing a bare $I$ is convenient only when the needed order is clear from context.

In entry notation, the identity uses the Kronecker delta: its entry is one when the row and column indices are equal and zero otherwise. The main diagonal runs from the upper-left corner to the lower-right corner. An anti-diagonal of ones defines a different permutation matrix, not the identity.

Worked example with a two-by-three matrix

Let $A$ have two rows and three columns. Multiply $I_2$ on the left. Each output row selects the matching row of $A$. Multiply $I_3$ on the right. Each output column selects the matching column of $A$. In both cases every entry remains in place.

The products $I_{3}A$ and $A{I}_2$ are not merely unequal; they are undefined because their inner dimensions do not match. Dimension checking should happen before any entry-by-entry multiplication.

Properties follow from its neutral role

For every nonnegative integer $k$, $I_{n^k}=I_n$. Its determinant is 1 and its trace is $n$. It is its own transpose and its own inverse. All of its eigenvalues are one because every vector keeps its direction and magnitude. The only identity that also acts as the zero matrix is the order-zero object in specialized conventions, not an ordinary positive-order matrix.

The multiplicative identity is unique. If two compatible matrices $E$ and $F$ both leave every matrix unchanged, then $E=EF=F$. This short argument explains why a matrix algebra does not have several competing identities of the same order.

Why identity matrices appear in elimination and transformations

An augmented matrix $[A\mid I]$ is used in Gauss-Jordan elimination because row operations that turn $A$ into $I$ apply the same transformations to the right block. If the process succeeds, that right block becomes $A^{-1}$. In a system $Ax=b$, reaching $Ix=A^{-1}b$ exposes the solution vector.

In geometry, $I_2$ maps each point $(x,y)$ to itself. Other matrices stretch, rotate, reflect or shear the plane. The identity is the baseline against which those transformations and their reversals are compared. Continue with the inverse matrix to see when a transformation can be undone.