Understanding Matrix Notation

A matrix is a rectangular array of numbers arranged in rows and columns. It is used to represent linear equations, transformations and vectors.
The position of each number in a matrix is denoted by its row number and column number.
The dimensions of a matrix are given as ‘rows’ by ‘columns’. For example, a ‘2 by 3’ matrix has 2 rows and 3 columns.

Matrices Elements

Every number in a matrix is called an element.
In a that a matrix has ‘m’ rows and ‘n’ columns, it is known as an ‘m x n’ matrix.
The elements of a matrix are usually denoted by lowercase letters with two subscript indices. For instance, the entry in the ‘i’-th row and ‘j’-th column of matrix ‘A’ is denoted by ‘a_ij’.

A row matrix is a matrix that has a single row, while a column matrix has just one column.
A matrix where the number of rows matches the number of columns is known as a square matrix.
A zero or null matrix is a matrix in which all elements are zero.
A diagonal matrix is a square matrix where all non-diagonal elements are zero.
A scalar matrix is a diagonal matrix where all diagonal elements have the same value.
A unit or identity matrix is a diagonal matrix where all diagonal elements are ‘1’ and it has the property that when it is multiplied by any matrix, the result remains unchanged.

Two matrices can be added or subtracted if and only if they have the same dimensions. In addition or subtraction, the corresponding elements are added or subtracted.
The result is a new matrix that has the same dimensions as the original matrices.

In matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second.
The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
The elements in the resulting matrix are obtained by computing the sum of the products of the corresponding elements in row ‘i’ of first matrix and column ‘j’ of the second one.

The transpose of a matrix is obtained by interchanging rows and columns.
For matrix ‘A’, its transpose is represented by ‘A^T’.
For a square matrix, if its transpose equals the original matrix, it is called a symmetric matrix.

The inverse of a matrix ‘A’ is denoted by ‘A^-1’, and it’s the matrix such that when ‘A’ is multiplied by ‘A^-1’, the result is the identity matrix.
Not all matrices have an inverse. Matrices that do not have an inverse are known as singular or degenerate.

Matrices can be used to represent and solve systems of linear equations.
An equation that consists of matrices is only valid if the size of each side of the equation (i.e., the number of rows and columns) is the same.