3 – MATRICES AND DETERMINANTS 1

MATRIX:

Every Matrix is denoted by Capital alphabet i.e. A,B,C etc

ORDER OF A MATRIX
It is the number of rows followed by the number of columns. e.g. if a matrix A has m rows and n
columns then its order will be m×n.
ROW MATRIX
A matrix having order 1×n is called row matrix or row vector. e.g. [1 2 3] is a row vector.
COLUMN MATRIX

A matrix having order m×1 is called column matrix or column vector. e.g. is a column vector.

RECTANGULAR MATRIX
If the number of rows and columns of a matrix is not equal, it is called rectangular matrix. e.g.

is a rectangular matrix.

SQUARE MATRIX

If the number of rows and columns of a matrix is equal, it is called square matrix. e.g. is a

square matrix.
DIAGONAL MATRIX
A square matrix is called diagonal if all the elements except the diagonal elements are zero,

some of the diagonal elements may be zero but not all. e.g. are

diagonal matrices.
SCALAR MATRIX
A square matrix is said to be scalar if all the elements except the principal diagonal are zero and

the diagonal elements are same. e.g. is a scalar matrix.

UNIT OR IDENTITY MATRIX

Prepared by: Inayat Khan Lecturer in Mathematics Edwardes College Peshawar

 3 – MATRICES AND DETERMINANTS 2

A square matrix is said to be identity matrix if all the elements except the principal diagonal

elements are zero and the diagonal element is “1”. e.g. is the identity matrix.

NULL OR ZERO MATRIX

A square matrix is said to be null or zero if all the entries in the matrix are zero. e.g.

is a null matrix.
ALGEBRA OF MATRICES
ADDITION AND SUBTRACTION
These operations are possible if both the matrices have same order.
SCALAR MULTIPLICATION
Let A be a matrix and “k” be any constant number, then kA is called the scalar
multiplication and “k” will by multiplied with all the elements of A.
TRANSPOSE OF A MATRIX
The transpose of a matrix A having order mxn is a matrix denoted by At having order
nxm. i.e. Transpose of a matrix is obtained by interchanging rows and columns.

e.g. if then

DETERMINANT OF A MATRIX
The determinant of a Matrix A over real numbers, is a real number denoted by |A|.

e.g. if then |A| = ad – bc.

SINGULAR AND NON-SINGULAR MATRICES

Let A be a square matrix. if |A| = 0, then A is called singular matrix otherwise non –
singular.
ADJOINT OF A SQUARE MATRIX
Let A be a square matrix and “C” denotes the cofactor matrix for “A”, then adjoint of A
denoted by adjA is given by: adjA = Ct

INVERSE OF SQUARE MATRIX

Let A and B be two square matrices. They are said to be inverses of each other if

Prepared by: Inayat Khan Lecturer in Mathematics Edwardes College Peshawar

 3 – MATRICES AND DETERMINANTS 3

AB = BA = I. Formula:

DIFFERENCE BETWEEN MINOR AND COFACTOR

Let ,then

Minor of 2 = and

Cofactor of 2 = etc

PROPERTIES OF DETERMINANTS

1. If any row or column of a determinant is zero, its value will be zero.e.g.

=0(because R1 = 0) and =0(Because C2 = 0)

2. If any two rows or columns of a determinant are identical (same), its value will be zero.e.g.

(Because R1 = R2) and =0(Because C1 = C3)

3. Interchanging any two rows or columns of a determinant changes the sign of determinant.e.g.

(By R12)

4. |A| = |A|t
5. If every element of a row or column of a square matrix A is multiplied by a real number k then

the determinant will be k|A|. e.g. .

6. If every element of a row or column of a square matrix A is the sum of two terms, then its
determinant can be written as the sum of two determinants.e.g.

UPPER TRIANGULAR MATRIX(U.T.M)

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 3 – MATRICES AND DETERMINANTS 4

If all the elements below the principal diagonal of a square matrix A are zero, it is called U.T.M.

e.g. .

LOWER TRIANGULAR MATRIX (L.T.M)

If all the elements above the principal diagonal of a square matrix A are zero, it is called L.T.M.

e.g. .

SYMMETRIC MATRIX
If At = A, then A is called symmetric matrix.
SKEW SYMMETRIC MATRIX
If At = - A, then A is called skew symmetric matrix.
HERMITIAN MATRIX
If , then A is called Hermitian matrix.
SKEW HERMITIAN MATRIX
If , then A is called skew Hermitian matrix.
ELEMETARY ROW OPERATION(E.R.O)
The following three operations performed on matrices are called E.R.O.
Interchanging of any two rows
Multiplication of a row by any non – zero scalar.
Addition of one row to another row.
ELEMENTARY COLUMN OPERATION (E.C.O)
The above three operations when performed on columns of a matrix are called E.C.O.
ECHELON FORM OF A MATRIX
An mxn matrix A is said to be in Echelon form if:
1. In each successive non – zero row, the number of zeros before the first non – zero entry
of a row increases row by row.
2. Every non – zero row in A preseeds every zero row.(If there is any) e.g.

are in Echelon form.

REDUCED ECHELON FORM OF A MATRIX

An mxn matrix A is said to be in Reduced Echelon form if:
1. It is in Echelon form.
2. The 1st non – zero entry in Rj lies in Cj is “1” and all other entries of Cj are zero.

Prepared by: Inayat Khan Lecturer in Mathematics Edwardes College Peshawar

 3 – MATRICES AND DETERMINANTS 5

e.g. are in Reduced Echelon form.

RANK OF A MATRIX
The number of non – zero rows of a matrix in its Echelon form is called Rank of that matrix.e.g.

the rank of is “2” and “1” respectively.

SULUTION OF SYSTEM OF LINEAR EQUATIONS

The system of equations AX = B has a unique solution, if |A| ≠ 0 and no solution if |A| = 0.
For the system of Homogeneous equations AX = 0, (0,0,0) is called trivial solution, solutions other
than (0,0,0) are called non – trivial solutions.
The system of Homogeneous equations AX = 0 has a trivial solution if the co- efficient matrix “A”
is non – singular i.e. |A| ≠ 0 and non – trivial solution if |A| = 0.

Prepared by: Inayat Khan Lecturer in Mathematics Edwardes College Peshawar