An m × n matrix is a rectangular array of m rows and n columns of scalars. Matrices are denoted A, B, etc., with entry a_ij in row i and column j. Special types include square matrices (m = n), diagonal, identity, triangular, and symmetric matrices. Matrices represent linear systems and transformations.
A matrix is simply a way of organizing numbers into a grid. An m × n matrix has m rows and n columns, so a 3 × 2 matrix has 3 rows and 2 columns. The entry in row i and column j is written a_ij — the row index always comes first. This double-subscript notation is the key to reading and writing matrix entries fluently: a_23 means row 2, column 3.
The power of the matrix format is that it packages a lot of information in a structured way that supports systematic computation. A system of two equations with three unknowns, for instance, can be represented as a 2 × 3 matrix of coefficients — the entire system fits in one object, and operations on the system become operations on the matrix. This is why matrices are the natural language for linear algebra: they translate between geometric transformations and numerical operations.
Several special matrix types appear constantly and deserve careful attention. A square matrix has equal numbers of rows and columns (m = n). The identity matrix I is a square matrix with 1s on the main diagonal and 0s everywhere else — it behaves like the number 1 in multiplication, leaving any matrix unchanged. A diagonal matrix has nonzero entries only on the main diagonal; these are the easiest matrices to work with because their properties are determined entirely by those diagonal entries. Triangular matrices (upper or lower) have zeros either below or above the main diagonal, which makes solving linear systems especially straightforward. A symmetric matrix satisfies A = Aᵀ, meaning it equals its own transpose — row i equals column i. Symmetric matrices arise throughout applied mathematics and have especially clean spectral properties.
Getting comfortable with the notation early pays large dividends. When you see a_ij, ask: which row? which column? When you see "m × n," remind yourself which dimension is which. The notation is designed to be consistent: matrix A is described as m × n (rows × columns), entry a_ij is (row i, column j), and this pattern never changes throughout linear algebra, regardless of the operation. Every computation you will do with matrices — addition, multiplication, inversion, decomposition — builds on fluency with this foundational bookkeeping.
This is a foundational topic with no prerequisites.
No prerequisites — this is a starting point.