In math, a matrix is a function that maps between two vector spaces. A matrix can also be thought of as a shorthand way to write a system of linear equations. For example, the matrix shown below takes in a vector of size 3 as input and produces a vector of size 2 as output.

Note, the variables represent the coefficients that define the function. The number of columns of the matrix corresponds with the size of the input vector and the number of rows corresponds with the size of the output vector. The same matrix can be written as a system of linear equations.

To express the input and output of the matrix the input vector is placed to the right of the matrix and then the output vector is the result of applying the matrix to .

The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix and the linear map it represents, such as whether the matrix is invertible or the system of equations it represents has a unique solution.

A vector is a mathematical object represented by an ordered list of numbers called components, which define its magnitude and direction in a multi-dimensional space. Vectors can be added, subtracted, and scaled by a scalar, following specific rules.