A matrix $A$ is said to be of dimension (shape) $m\times n$ if it is an $m\times n$ matrix.

Let us consider a few examples.

Warning: the following $\left(\begin{array}{ccc}1& 2& 3\\ 0& & -2\end{array}\right)$ is not a matrix.

It is important to distinguish matrices, equivalently to identify matrices that are the same.

Any matrix is equal to itself. From the above examples $U=U$ and $V=V$, however $U\ne V$. Recall the matrix $V=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\\ 0& -1& -2\end{array}\right)$ we have

and

There are multiple special matrix classes that have their own names as they are encountered quite often. For example $1\times m$ matrix is a row matrix sometimes just called a row or a row vector e.g., $\left(\begin{array}{ccccc}1& 3& 3& 2& 1\end{array}\right)$, $\left(\begin{array}{ccccc}3& 9& -6& 4& 3\end{array}\right)$ and $\left(\begin{array}{ccccc}2& 6& -4& 2& 2\end{array}\right)$ are all $1\times 5$ row matrices. These are the rows of matrix $𝒰$ defined above.

Similarly $n\times 1$ matrix is a column also called a column matrix or a (column) vector. Each of the following

is a $\mathrm{3<mo>\times </mo><mi>1</mi>}$ matrix, equivalently a column. They are respectively the second, third, forth and fifth column of matrix $U$. The last column matrix is also the first column of $U$.

The entries of a row/column matrices are also called components.

Here are two square matrices.

In a matrix $ℒ={\left\{l_{ij}\right\}}_{1\le i\le m,1\le j\le n}$ the set of entries $l_{ii}$ is called the (main) diagonal. Often matrices have only zeroes either above or below their diagonal. In that case they are called triangular matrices.

A square matrix whose only non-zero entries are on its main diagonal is called diagonal matrix. If all diagonal entries are equal to each other the matrix is scalar matrix. A scalar matrix whose diagonal entries are all equal to one is an identity matrix. There is only one identity matrix of a given order.