What is a Matrix?

Linear Algebra (Matrices) · Axiom Academy

A matrix is a rectangular array of numbers — learn the formal notation A = [a_ ij ] , its dimensions, and how to address any entry. 1. From Scattered Data to a Rectangular Array Suppose you are tracking three students across four tests. Listed loose, the scores are hard to work with. Snap them into a grid — one row per student, one column per test, wrapped in square brackets — and you have a matrix A . Each row is one student; each column is one test 2. Dimensions: Count the Rows, Count the Columns The size (or order ) of a matrix is written , always rows first, columns second . Count down the left to get m , count across the top to get n . For our matrix A that is — and total entries. The number of horizontal rows. Here m = 3 : three students. The number of vertical columns. Here n = 4 : four tests. Always state it in that order. A matrix is not the same shape as a one. Multiply the two: numbers fill the array. A single horizontal slice is a row vector — row 1 of A is . A single vertical slice is a column vector — column 3 of A is . A matrix is just these vectors stacked together. 3. The Address a_ ij : Row i , Column j To name one specific number we use subscripts: a_ ij is the entry in row i , column j — row first, column second, like coordinates on a grid. Watch the row i and column j slide in; where they cross is exactly a_ ij . First subscript i = the row number . Second subscript j = the column number . So in A : a_ 12 = 5 (row 1, col 2) and a_ 23 = 9 (row 2, col 3).

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.