The language of matrices

Basics of Matrices

In general, a matrix is a rectangular array of numbers.
The order of a matrix is given by the number of rows and columns it has. A 2x3 matrix has 2 rows and 3 columns.
A vector can be regarded as a matrix with only one column or one row.
A square matrix is a matrix where the number of rows is equal to the number of columns.
The numbers in a matrix are commonly referred to as elements or entries.

Row matrix: Matrix with a single row.
Column matrix: Matrix with a single column.
Zero or Null Matrix: All elements in the matrix are zero.
Diagonal Matrix: Elements outside the main diagonal are zero.
Scalar Matrix: All non-diagonal elements are zero and all diagonal elements are same.
Unity or Identity Matrix: All diagonal elements are one and non-diagonal elements are zero.

Addition and Subtraction: Same order matrices can be added or subtracted element by element.
Scalar Multiplication: Every element of matrix can be multiplied by a scalar.
Matrix Multiplication: The number of columns in the first matrix should be equal to the number of rows in the second matrix for multiplication. The multiplying process refers to the dot product.
Transpose of a Matrix: Transpose of a matrix is a new matrix whose rows are the columns of the original.

Main Diagonal: In a square matrix, the main diagonal (or principal diagonal) is the set of elements running from the top left corner to the bottom right.
Determinant: A special numerical value calculated from a square matrix.
Minor and Cofactor: For each element in the matrix, there is a corresponding minor and cofactor derived based on determinant of sub-matrix.
Inverse of a matrix: Only square matrices have inverses. The matrix when multiplied by its inverse results in identity matrix. An inverse matrix does not exist if determinant is zero (matrix is called singular).
Rank of a matrix: The maximum number of linearly independent rows in a matrix.