• recognise a square matrix and its order;
• find the main diagonal and compute the trace;
• recognise diagonal, identity and zero matrices;
• recognise upper and lower triangular matrices.

Some matrices have a shape, or a pattern of zeros, that makes them far easier to work with than the rest. This lesson names the ones the whole course keeps meeting.

A matrix with as many rows as columns is called square. A square matrix with $n$ rows and $n$ columns has order $n$.

This matrix is $3\times3$, so it is square of order $3$. A $2\times3$ matrix is not square: it has more columns than rows.

The entries whose row index equals their column index — $a_{11}$, $a_{22}$, $a_{33}$ and so on — lie on the main diagonal. In a square matrix it runs from the top left corner to the bottom right one.

Here the main diagonal holds $4$, $5$ and $9$. The other corner-to-corner line, from the top right to the bottom left, is not the main diagonal.

The trace of a square matrix is the sum of the entries on its main diagonal. It is written $\operatorname{tr}A$.

A square matrix is diagonal when every entry off the main diagonal is zero. The entries on the diagonal may be anything at all, zero included.

This $D$ is diagonal even though one diagonal entry is $0$: the condition is on the entries off the diagonal. It is often written $\operatorname{diag}(3,-2,0)$.

The identity matrix of order $n$, written $I_n$ or just $I$, is the diagonal matrix whose diagonal entries are all $1$.

Its entries follow one rule: $1$ where the row index equals the column index, and $0$ everywhere else. Later in this chapter it will turn out to play the part the number $1$ plays in multiplication.

A matrix whose entries are all zero is a zero matrix, written $O$. There is one of every size, square or not.

A square matrix is upper triangular when every entry below the main diagonal is zero, and lower triangular when every entry above it is zero.

In index form, $U$ is upper triangular when $u_{ij}=0$ whenever $i>j$, and is lower triangular when whenever . The name says where the non-zero entries may live.

• A matrix is square when it has as many rows as columns; an $n\times n$ matrix has order $n$.
• The main diagonal holds $a_{11},a_{22},\ldots,a_{nn}$, and the trace is their sum.

Great work! You now know how to:

• recognise a square matrix and its order;
• find the main diagonal and compute the trace;
• recognise diagonal, identity and zero matrices;
• tell upper triangular from lower triangular.