Matrices and determinants

A matrix is a set of numbers arranged in rows and columns. Example:

Row matrix: Has only one row.

Column matrix: Has only 1 column.

Square matrix: Has the same number of rows as columns.

Zero matrix: All elements are 0.

Triangular matrix: The elements above or below its main diagonal are 0.

Diagonal matrix: The elements above and below its main diagonal are 0. It must be a square matrix.

Identity or unit matrix: A diagonal matrix where all the elements of its main diagonal are 1.

Transpose matrix: Given a matrix $A$, the transpose is the matrix $A^T$ obtained by swapping rows and columns.

Symmetric matrix: A square matrix equal to its transpose:

It is denoted by rank A. It is the number of linearly independent lines (rows or columns), or also:
It is the dimension of the largest non-zero minor.
For a matrix A with dimensions $m \times n$, its rank cannot be greater than the smallest dimension of A.

Calculating the rank by the Gauss-Jordan method.
In general, it consists of making the maximum number of lines zero by applying elementary row operations, and the rank will be the number of non-zero rows:
Elementary operations are:

• Multiplying a row by a non-zero number.
• Adding (or subtracting) to a row the multiple of another row.
• Swapping the order of the rows.

Calculating the rank using determinants.
It consists of finding non-zero minors of the highest possible order, meaning, starting with the "largest" minors of A, and reducing the size until finding one that is non-zero.

The order or dimension of a matrix indicates the number of rows and number of columns. It is denoted as $m \times n$ ($m$ being the number of rows, and $n$ the number of columns). E.g.:

If the matrix is square, it can be called a square matrix of order $n$, where $n$ is the number of rows and columns.

Matrix addition: They must have the same dimensions. It is obtained by adding the elements in the same position (element-wise addition).

Product of a real number and a matrix: A matrix of the same order is obtained, in which each element is multiplied by said real number.

Matrix multiplication: Two matrices A and B can be multiplied if the number of columns in A equals the number of rows in B. We obtain a matrix that has the same number of rows as A and the same number of columns as B. The element $c_{ij}$ of the product matrix is obtained by multiplying each element of row $i$ of matrix A by its corresponding element of column $j$ of matrix B and adding them. Example:

Definition: Let A be a square matrix. The inverse matrix of A, denoted by $A^{-1}$, satisfies:

Where $I$ is the identity matrix of the same dimension as A.
Matrices that have an inverse are called regular or invertible. Otherwise, they are called singular or degenerate.
A matrix is invertible if its determinant is non-zero.

Every square matrix is associated with a real number called its determinant, and it is denoted by $\det A = |A|$

Determinant of a 3 x 3 matrix:
We expand it by copying the first two columns and adding them at the end. We perform the calculation indicated in the figure:

Determinant of a 4 x 4 matrix:
Choose a row or column with the greatest possible number of zeros. We multiply each element by its cofactor.
Ex.: finding the determinant by cofactors of the 2nd column

The Minor of a matrix is the determinant of any $k \times k$ submatrix.

The Cofactor of an element $a_{ij}$ is the value of the determinant of order $n-1$ obtained by deleting row $i$ and column $j$ from the matrix.

The Cofactor (with sign) of an element $a_{ij}$ of a matrix is its minor multiplied by $(-1)^{i+j}$.

The adjugate matrix, $A^*$, or matrix of cofactors, is the one in which each element is replaced by its cofactor.

Inverse of a 2 x 2 Matrix A 2 x 2 matrix has an easy-to-remember inverse according to the following formula: