1 Allosaurus: RAAARRRHH!!
1 Iki Piki: Quercus! Use your Matrix kung fu on the Allosaurus!
2 Quercus: An n×n matrix is invertible if and only if its determinant is non-zero!
2 Allosaurus: RAAARRRHH!?
3 Quercus: An n×n matrix has at least one and at most n distinct complex eigenvalues!
3 Allosaurus: Raaarrrhh!
4 Iki Piki: That's not quite what I meant.
4 Quercus: He's weakening!
4 Allosaurus: Raaarr...

A matrix is a rectangular two-dimensional array of numbers (or mathematical expressions).

Matrices may be used to represent certain physical quantities or properties that cannot adequately be described by either a single number or a vector. They are particularly useful for non-isotropic physical properties, that is, properties that vary depending on which direction you measure them in. For example, the elasticity of cloth, which varies parallel and perpendicular to the weave.

Like numbers and vectors, matrices have certain mathematical rules that govern how they can be added, multiplied, or other operations. For matrices of size n×n, that is n rows and n columns, there is a unique matrix I which, when multiplied by another other matrix A, produces the same matrix A as the answer: AI = IA = A^[1] This unique matrix I is kind of like the number 1 in arithmetic multiplication, and is called the identity matrix.

Now, in arithmetic, given any non-zero number x, you can find another number y such that xy = 1. The number y is the reciprocal, or inverse, of x: y = 1/x. For example, if x is 5, then y is 1/5, and 5×(1/5) = 1.

In matrix multiplication, by contrast, it is not guaranteed that there is a "reciprocal" or inverse matrix for any given non-zero matrix A. That is, for any given non-zero n×n matrix A there might be a matrix B such that the products AB and BA are equal to the identity matrix I, or there might not be such a matrix B.

To determine if A has an inverse matrix, that is, if the matrix A is invertible, you can calculate a special property of the matrix A, called the determinant. The determinant is a number produced by combining all of the numbers in the matrix in a specific way. If the determinant of A is non-zero, then an inverse of A exists - that is, the matrix A is invertible. On the other hand, if the determinant of A is zero, then an inverse of A does not exist - that is, the matrix A is not invertible.

Eigenvalues I've talked about before, so won't go into them again here. Except in that discussion I said that eigenvalues are properties of a function. Eigenvalues of a matrix are the eigenvalues of the function formed by using the matrix to multiply vectors.

[1] Matrix multiplication is non-commutative. This means that, unlike normal arithmetic multiplication, the order of the matrices in the multiplication is important. With normal numbers x and y, the products xy and yx are the same number. For example: 3×5 = 5×3 = 15.

By contrast, for two matrices A and B, in general the products AB and BA are different matrices.