At first glance a matrix is just a rectangular grid of numbers, written in square brackets:

But that grid is secretly a machine. Feed a matrix a vector and it hands back a new one — stretched, rotated, squashed or flipped. The very same idea, scaled up to enormous grids, is what rotates a photo on your phone, blurs a video call's background, tells a robot arm how to move its joints, and — in the form of one famously gigantic matrix — decides which web pages Google shows you first. A matrix isn't a table you read; it's an instruction you run. Before we can run one, though, we need to be fluent at reading one — so that's where we start.

A matrix's size is given as rows × columns. The matrix above is 2 \times 3 — two rows, three columns. We name the entry in row i, column j as a_{ij}; here a_{12} = -1 and a_{23} = 4. Row first, then column — always.

Reading the grid: rows, columns, dimensions

A row runs left to right; a column runs top to bottom. When we state a matrix's dimensions we always say rows first: an m \times n matrix has m rows and n columns, giving it m \times n entries in total, one for every row–column pairing. The entry a_{ij} lives where row i crosses column j — think of it like a seat number in a theatre: row number first, seat-along-the-row number second.

The lowercase letter matches the matrix's capital: entries of A are called a_{ij}, entries of a matrix B would be b_{ij}, and so on. Subscripts i and j are just placeholder names for "whichever row" and "whichever column" — swap in actual numbers, like a_{31}, and you're pointing at one specific entry. Rows are always counted starting from the top, columns always starting from the left, so a_{11} is forever the entry in the top-left corner.

Worked example 1: naming the size

Count rows top-to-bottom, then columns left-to-right, for each of these:

P has 2 rows and 2 columns: it's 2 \times 2. Q has 3 rows but only 1 column: 3 \times 1 — all column, no width, so we call it a column vector. R flips that: 1 row, 4 columns, so 1 \times 4 — a row vector. And S has 3 rows and 3 columns, 3 \times 3: equal rows and columns makes it a square matrix (this particular one, with 1s down the diagonal and 0s elsewhere, is special enough to have its own name later on). Notice how easy it is to get Q and R backwards if you don't count carefully — they hold the same four numbers arranged in opposite shapes.

Worked example 2: find the entry

Take the very first matrix again, A = \begin{bmatrix} 2 & -1 & 0 \\ 3 & 5 & 4 \end{bmatrix}, and read off a few entries by hand before touching anything. Row 1 is (2, -1, 0) and row 2 is (3, 5, 4). So a_{12} means row 1, column 2: scan along row 1 to the second slot, and you land on -1. Meanwhile a_{21} means row 2, column 1: scan along row 2 to the first slot, and you land on 3. Same two digits, "1" and "2" — swap their order and you land on a completely different number. That is exactly why the convention row first, then column has to be non-negotiable.

Now dial a row and a column below and watch the highlighted cell track your choice. Get used to the convention — countless bugs in real code come from swapping row and column indices.

Worked example 3: a matrix hiding in a mark sheet

You don't need square brackets to be looking at a matrix — any grid of related numbers qualifies. A teacher's mark sheet for three students across three subjects is a genuine 3 \times 3 matrix S, students as rows and subjects as columns:

Number the rows Ann, Ben, Cara (1, 2, 3) and the columns Maths, Science, Art (1, 2, 3). Ben's Science score is entry a_{22} — row 2 (Ben), column 2 (Science) — which the table shows is 88. Try it yourself: what is a_{33}, Cara's Art score? Row 3 is Cara, column 3 is Art, and the table reads 84. Spreadsheets, databases and grade books are matrices wearing everyday clothes.

Notice the choice of "students as rows, subjects as columns" was ours to make — we could just as easily have flipped it, with subjects as rows and students as columns. Nothing about the numbers forces one layout over the other. What matters is picking a convention and being explicit about it, because a_{23} in the students-as-rows layout is a completely different score from a_{23} in the subjects-as-rows layout, even though it's the exact same table of marks underneath.

Shape decides what a matrix can do

Two special shapes matter most. A matrix with one column is exactly a column vector — so vectors are just skinny matrices — and a matrix with one row is a row vector, its sideways twin. A square matrix has as many rows as columns, and those are the ones that can act as transformations of a space onto itself: a 2 \times 2 matrix can rotate or stretch the 2-D plane, sending every point to a new point in that same plane. A non-square matrix, like our 2 \times 3 example, instead moves between spaces of different sizes altogether. Shape decides what a matrix is allowed to do — long before we ever multiply anything.

See it explained