Understanding Matrices

Matrices are rectangular arrays of numbers organized in rows and columns. They are especially useful for solving systems of linear equations.
The order of a matrix is described by the number of rows and columns it has, in that order. A matrix with ‘m’ rows and ‘n’ columns is an ‘m x n’ matrix.
A matrix with the same number of rows and columns is a square matrix.

Operations with Matrices

Matrices can be added or subtracted by adding or subtracting the corresponding elements.
Matrices can be multiplied by a scalar (simple number) by multiplying each element by that scalar.

Matrices multiplication involves multiplying the rows of the first matrix by the columns of the second matrix. This can only occur if the number of columns in the first matrix is equal to the number of rows in the second.
It’s important to remember that matrix multiplication is not commutative. Switching the order of the matrices (AB vs BA) can result in different outcomes.

The determinant is a special number that is calculated from a square matrix. The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, calculus, and more.
The inverse of a matrix A (denoted as A^-1) is a special type of matrix. When a matrix is multiplied by its inverse, the result is the identity matrix.
Not all matrices have inverses. Matrices that do not have an inverse are known as singular matrices.

Matrices are used in various fields such as physics, computer graphics, probability theory, statistics, calculus, numerical analysis, and more.
In the context of Pure Mathematics, matrices can serve to draw real world problems in their matrix form and then execute mathematical operations.