A matrix is a rectangular array of numbers or functions which will be enclosed in brackets. Matrix has rows, horizontal lines of entries and columns, vertical lines of entries. A matrix of any size m x n is called a rectangular matrix.
Square matrix, has as many rows as columns. Matrices of different sizes cannot be added. Matrix addition is commutative and associative.
Matrix having just a single row or column is called vectors. Row vector has one row and Column vector has one Column.
Matrix Multiplication
Transposition
The transpose of a matrix is obtained by writing its rows as columns (or equivalently its columns as rows). Transposition converts row vectors to column vectors and vice versa.
Rules of Transposition
Symmetric Matrix
Square matrix whose transpose equals to the matrix itself.
Skew Symmetric Matrix
Square matrix whose transpose equals to the minus of matrix.
Upper Triangle Matrix
Square matrix having non zero entries on and above the main diagonal.
Lower Triangle Matrix
Square matrix having non zero entries on and below the main diagonal.
Diagonal Matrix
Square matrix having non zero entries only on the main diagonal.
Unit Matrix
Diagonal matrix having all its main diagonal element are 1.
Stochastic Matrix
Square matrix with all entries non negative and all columns sums equal to 1.
Matrices are used to solve systems of linear equations. The approach to solving linear systems is called the Gauss elimination method.
The system is called linear because each variable x appears in the first power only, just as in the equation of a straight line.
x - variable ; a - coefficient ; b - numbers
If all the b are zero, then the system is called a homogeneous system. If at least one b is not zero, then the system is called a nonhomogeneous system.