**A square matrix is a very basic matrix typology that is characterized by having the same order of both rows and columns.**

In other words, a square matrix has the same number of rows (n) and the same number of columns (m).

## Representation of a square matrix

We can create infinite combinations of square matrices as long as we respect the restriction that the number of columns and rows must be the same.

## Square matrix of order n

Since in a square matrix the number of rows (n) is equal to the number of columns (m), we mathematically say that n = m.

Then, starting from this equality, it is enough to only indicate the number of rows (n) that the matrix has.

Why? Well, because knowing the number of rows (n) we will also know the number of columns (m) since n = m.

The order tells us the number of rows (n) and columns (m) that a matrix has. In the case of the square matrix, just by indicating the order of the rows (n) we will already know the order of the columns (m). So when we are told that a square matrix is of order n, it means that this matrix has n rows and n columns given that n = m and m = n.

## Differentiate a square matrix from other non-square matrices

How can we remember that a square matrix has the same number of rows and columns?

Let’s think of a square. That is, squares are famous for having sides of the same length. So a square matrix will also have this characteristic: the number of rows and columns will match.

Apart from the analytical vision, from the geometric vision, a square matrix will also look like a square:

Matrix A: square shape => Square matrix.

Matrix B: rectangle shape => Non-square matrix.

Matrix C: rectangle shape => Non-square matrix.

## Applications

The square matrix is the basis for many other types of matrices such as the identity matrix, the triangular matrix, the inverse matrix, and the symmetric matrix. In addition, it is also the basis for complex operations such as the Cholesky decomposition or the LU decomposition, both widely used in finance.

The use of matrices in econometrics greatly facilitates calculations when linear regressions are multiple linear regressions. In these cases, all the variables and coefficients can be expressed in matrix form and help in understanding the study.

## Theoretical example

Square matrix of order 2: 2 rows and 2 columns.

Square matrix of order 3: 3 rows and 3 columns.

Square matrix of order n: n rows and n columns (n = m):

Matrix division