Matrices, from zero to ohhh.

A matrix is just a rectangular grid of numbers, arranged in rows and columns. That's it. The magic isn't in the object — it's in what it does: it transforms, encodes, and solves.

Think of a matrix as a machine. You feed it a vector (a column of numbers), and out comes another vector — usually a transformed version of the first one.

A 2×3 matrix has 2 rows and 3 columns:

The plural of "matrix" is matrices (pronounced MAY-trih-seez). The singular adjective is matrix.

We describe a matrix by its dimensions: m × n means m rows and n columns. An entry is named a_ij where i is the row and j is the column.

Special cases worth knowing:

• Row vector: 1 × n — a single row of numbers.
• Column vector: n × 1 — a single column of numbers.
• Square matrix: n × n — same number of rows and columns. These are the stars of the show.
• Identity matrix I_n: square with 1s on the diagonal and 0s elsewhere.
• Zero matrix: every entry is 0.

The two simplest operations, and the rules are dead simple:

Matrix addition. Add two matrices of the same shape, entry by entry:

Scalar multiplication. Multiply every entry by the same number (the "scalar"):

Multiplication is where matrices earn their keep. The rule:

To compute the entry (AB)_ij, take the dot product of row i of A with column j of B:

The shape rule. If A is m × n and B is n × p, then AB is m × p. The inner dimensions must match.

The transpose of A, written A^T, flips the matrix along its diagonal — rows become columns.

A matrix that equals its own transpose is called symmetric. These pop up everywhere in physics and statistics.

Handy identities:

• (A^T)^T = A
• (A + B)^T = A^T + B^T
• (AB)^T = B^TA^T (note the flip!)

The determinant, written det(A) or |A|, is a single number that captures a square matrix's character.

For a 2×2 matrix, the formula is delightfully simple:

For a 3×3 matrix, you use the rule of Sarrus or cofactor expansion. The idea: you recursively expand along a row.

The inverse of A, written A⁻¹, is the matrix that "undoes" A. It's the matrix equivalent of a reciprocal.

For a 2×2 matrix:

The classic use case: you have a system of linear equations, and you want to solve it fast.

For example, suppose:

• 2x + 3y = 8
• x − y = 1

That becomes:

The solution (x = 11/5, y = 6/5) is the unique point where two lines intersect — and that's a geometric view you'll see a lot of in Linear Algebra.

This is the punchline of a lot of linear algebra. Some vectors are special: when a matrix transforms them, they don't change direction — only magnitude.

v is an eigenvector, λ (lambda) is its eigenvalue. The matrix acts like a simple scalar on these special vectors.

To find them, you solve:

That's the characteristic polynomial. For a 2×2, it's a quadratic — two eigenvalues. For 3×3, a cubic. And so on.

That's the end of the matrix tutorial! If you want to keep going, the next stop is Linear Algebra — which takes all of this and runs with the geometry.