Examples of 3×3 Matrices With Common Properties

The phrase “meet properties” does not identify which properties are required. The examples below cover several common conditions on real $3\times3$ matrices, so you can select the ones that fit your problem. Each example includes a check that shows why it works.

If you have a specific list of conditions, apply them together: a matrix satisfying each condition separately does not necessarily satisfy all of them at once.

Common examples

Here are useful examples, written by rows. All entries are real.

Property	Example	Quick check
Identity	$I=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$	$IA=AI=A$ for every $3\times3$ matrix $A$.
Symmetric	$S=\begin{pmatrix}1&2&0\\2&3&-1\\0&-1&4\end{pmatrix}$	$S^T=S$.
Skew-symmetric	$K=\begin{pmatrix}0&1&-2\\-1&0&3\\2&-3&0\end{pmatrix}$	$K^T=-K$.
Invertible	$A=\begin{pmatrix}1&2&0\\0&3&1\\0&0&-2\end{pmatrix}$	It is triangular and $\det(A)=1\cdot3\cdot(-2)=-6\ne0$.
Singular, rank two	$B=\begin{pmatrix}1&0&0\\0&1&0\\1&1&0\end{pmatrix}$	The third column is zero, and the first two columns are independent; hence $\operatorname{rank}(B)=2$.
Idempotent	$P=\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}$	$P^2=P$.
Nilpotent	$N=\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}$	$N^3=0$, although $N^2\ne0$.
Orthogonal	$Q=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&1\end{pmatrix}$	$Q^TQ=I$.

How to verify a property

Choose the test that matches the definition. To check symmetry, transpose the matrix and compare it with the original. To check skew-symmetry, compare the transpose with the negative of the matrix. For invertibility, a nonzero determinant is a quick test; for triangular matrices, multiply the diagonal entries to obtain the determinant.

For rank, count the independent rows or columns after row reduction. For idempotence, multiply the matrix by itself. For nilpotence, keep taking powers until a power is zero. For orthogonality, verify $Q^TQ=I$; equivalently, the columns of $Q$ are mutually perpendicular unit vectors.

Worked check: an idempotent matrix

For $P=\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}$, direct multiplication gives

$$P^2=\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix} \begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix} =\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}=P.$$

So $P$ is idempotent. It is also singular, since its determinant is zero. In fact, its rank is two. This illustrates that one matrix can have several properties at once.

A pair of matrices that do not commute

If you need matrices $A$ and $B$ for which $AB\ne BA$, use

$$A=\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix},\qquad B=\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}.$$

Multiplication gives $AB=\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}$, while $BA=\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}$. Thus $AB\ne BA$. Matrix multiplication is not commutative in general, even for square matrices of the same size.

Common mistakes and incompatible conditions

• Checking only one condition: If a task asks for a symmetric, invertible matrix, the example must satisfy both requirements. The symmetric matrix in the table is not automatically a suitable example for every other property.
• Confusing singularity with rank zero: A singular $3\times3$ matrix has rank less than three, but it can still have rank one or two. The zero matrix has rank zero.
• Assuming a nilpotent matrix is invertible: A nilpotent matrix has determinant zero, so it cannot be invertible. For example, $N$ above is nonzero but $N^3=0$.
• Overlooking special cases: A matrix that is both symmetric and skew-symmetric over the real numbers must be zero. Indeed, $A^T=A$ and $A^T=-A$ imply $A=-A$, hence $A=0$.
• Treating all idempotent matrices as invertible: An invertible idempotent matrix must be the identity. If $P^2=P$ and $P$ is invertible, multiply by $P^{-1}$ to get $P=I$.

These checks help distinguish properties that can be combined from conditions that rule one another out.