- A set of mn elements arranged in a rectangular arrangement along m rows and n columns enclosed by the brackets [ ] or ( ) is called m by n matrix.
Order of matrix
- The order of a matrix is a number of rows and columns of the matrix.
- From above example the order of matrix ot has no of rows=2 and no of columns=4.
- The Matrix has a single row is called a row matrix.
- The matrix has a single column is called a column matrix.
- A matrix having an equal number of rows and columns is called a square matrix.
- In a matrix, if all the elements are zero then that matrix is called a null or zero matrices and it is always denoted by 0.
- In a square matrix, all the elements except the elements in the main diagonal are zeros, then the matrix is called a diagonal matrix.
- A square matrix in which all the elements of its leading diagonal are equal and the other elements all the zero is called a scalar matrix.
- A diagonal matrix that has unity in the leading diagonal is called a unit matrix. It is always denoted by I.
Upper triangular matrix
- A square matrix in which all elements below the leading diagonal zero is called an upper triangular matrix.
Lower triangular matrix
- A square matrix in which all elements above the leading diagonal are zero is called a lower triangular matrix.
Transpose of a matrix
- The matrix got from the given matrix A, by interchanging the rows and columns is called the transpose of A and denoted by .
- A square matrix A is symmetric if .
Skew symmetric matrix
- In skew- symmetric A square matrix, A then skew-symmetric is .
- In a singular matrix, a square matrix A is said to be singular if the value of A is zero.
Equality of matrices
- Two matrices A and B is said to be equal if and only if
- Both matrices of the same order
- Each element A is equal to the corresponding element of B.
Addition of matrices
- Let A and B be two matrices of same orders, then their sum A+B is defined as the matrix where each element is the sum of the corresponding elements of A and B.
Subtraction of matrices
- Let A and B be two matrices of the same orders, then their difference A-B is defined as the matrix where each element is the difference of the corresponding elements of A and B.
- The product of a scalar matrix A by a scalar K is a matrix in which each element is K times the corresponding elements of A.
- Example: k= 2,
Multiplication of matrices
- Two matrices can be multiplied only when the number of columns in the first matrix is equal to the number of rows in the second matrix.
- E.g., if A = m×n, B = n×p then AB = m×p