What Are the Identity Matrix and Zero Matrix?

The identity matrix leaves another matrix unchanged under multiplication, while the zero matrix contains only zeros and leaves a matrix unchanged under addition. These are the matrix equivalents of the numbers $1$ and $0$, respectively.

Definitions and Properties

An identity matrix is a square matrix with $1$ on its leading diagonal and $0$ everywhere else. It is denoted by $I$, often with its order shown as a subscript:

$$I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad I_3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}.$$

If $A$ is an $m\times n$ matrix, then

$$I_mA=A\qquad\text{and}\qquad AI_n=A.$$

The identity matrices require different orders here because matrix multiplication depends on compatible dimensions.

A zero matrix, denoted by $O$, has every element equal to zero. Unlike an identity matrix, it does not have to be square. For example,

$$O_{2\times3}=\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}.$$

For example, if $A=\begin{pmatrix}2&-1\\4&3\end{pmatrix}$, then and .

How the Operations Work

Consider a $2\times2$ matrix $A$. Multiplying on the right by $I_2$ forms each output column from the corresponding column of $A$; multiplying on the left forms each output row from the corresponding row. In both cases, the diagonal ones select the original entries, while the off-diagonal zeros prevent unwanted combinations.

Addition with a zero matrix works entry by entry: $a_{ij}+0=a_{ij}$. By contrast, multiplying by a compatible zero matrix produces a zero matrix because every resulting entry is a sum of products containing zero.

Dimension checks determine which statement is valid. If $A$ is $2\times3$, then $I_2A=A$ and $A{I}_3=A$, but is undefined.

A common misconception is that $I$ and $O$ are ordinary numbers. They are matrices, so their dimensions must be checked before addition or multiplication.

Exam Technique

This is HL-only content from AHL 1.14: Introduction to matrices and may appear in any GDC-active paper style. In an IB response, write the required matrix explicitly, check its order, and distinguish carefully between $I$, $O$, and scalar values $1$ and $0$.