2. Only Square Matrices Can Have Determinants

DETERMINANTS

1. Determinant of a square matrix: Determinant of a square matrix is number or an expression associated to the matrix and calculated in a particular way

NOTE:

1. Determinant of a square matrix A is denoted as, where is not the modulus of A as the determinant can be negative.

2. Only square matrices can have determinants.

DETERMINANTS OF MATRICES ORDER 1 AND 2.

(i) Determinant of a square matrix of order 1 is the element of the matrix itself.

Example: If then

(ii) Determinant of a square matrices of order ‘2’ can be calculated as

follows:

Example:

MINOR AND CO-FACTOR OF ELEMENTS OF A SQUARE MATRIX

(i) Minor of an element of a square matrix: Minor of an element of a square matrix is defined as the determinant of the square matrix, which is obtained by deleting the row and column in which the element lies.

NOTE: Minor of an element of the square matrix is denoted by

Example:

1. In the square matrix A =the minors of all the elements are given as :

Similarly other minors can be found

(ii) Co-factors of the elements of a matrix: If is a square matrix of order ‘n’ then Co-factor of the elementis denoted as and given by

EXAMPLE:

Find the minors and co-factors of the element of the square matrix.

Solution:

Similarly the students are advised to find the remaining 7 minors and co-factors.

DETERMINANT OF A SQUARE MATRIX: Determinant of a square matrix is defined as sum of the product of the elements of any row (column) with their respective co-factors.

i.e. If is a square matrix of order ‘3’ then the determinant of A is given by:

(Along row 1)

(Along column 1)

Similarly, the determinant can be calculated along other row (column).

Example: Find the determinant of the matrix A=.

Solution:

(Along row 1)

Similarly we can find the determinant along any other row or column.

NOTE:

1. The determinant remains same along any row or column.

2.(i) For easier calculations, we shall expand the determinant along that row or columnwhich contains maximum number of zeros.

(ii) While expanding, instead of multiplying by j, we can multiply by +1 or –1 according as (i + j) is even or odd.

(iii) Signs of co-factors with respect to their minors in a square matrix.

PROPERTIES OF DETERMINANTS

1. Determinant of a square matrix and its transpose is same.

i.e. The value of a determinant remains unchanged if its rows and columns are interchanged.

i.e. If is a square matrix and is its transpose then .

2. If we interchange any two rows or columns of a determinant then the determinant changes its sign.

3. If all the elements of a row or column are all zeros then the value of the determinant will be zero.

4. If elements of any two rows or columns are identical (same) or proportional then the value of the determinant becomes zero.

Example

(i), because here and are proportional.

(ii), because here and are identical.

5. Scalar multiplication of determinant of a square matrix is equal to the determinant of the square matrix which is obtained by multiplying any one row (or column) by that scalar.

Conversely, If there is some element common in any one row (or column) of a determinant then it can be taken common from that determinant.

Example:

(i) ,here the scalar ‘k’ is multiplied to the elements of column 2.

(ii) ,here the number ‘2’ is taken common from elements of .

NOTE: If is a square matrix of order ‘n’ then.

6. If we add (subtract) the scalar multiple of the elements of any row (column) to the elements of some other row or column then the value of the determinant remains the same.

NOTE: If we add (or subtract) the scalar multiples of row to the elements of row then this operation can be symbolically written as (Read as row changes to k times the elements row). Similar is the operation.

Example:

Similar other operations can be done by making similar operations

7. If the elements of any row (or column) can be expressed as the sum of two or more elements then the whole determinant can be expressed as the sum of two or more determinant along that row or column.

AREA OF THE TRIANGLE FORMED BY THREE POINTS

Area of the triangle formed by three points whose co-ordinates are is given by:

Area of the triangle =.

NOTE: Three points will be collinear iff

ADJOINT OF A MATRIX: Adjoint of a matrix is transpose of the matrix whose elements are co-factors of the corresponding elements of the given square matrix.

i.e. If is any square matrix of order ‘’then its adjoint is:

Where, being the minor of the element.

RESULTS: If A and B are any two matrices of same order then

1.

2., whereI is the identity matrix.

3. Ifis a square matrix of order ‘n’ then.

4.

SINGULAR AND NON-SINGULAR MATRIX

SINGULAR MATRIX: A square matrix is said to singular iff .

NON-SINGUALR MATRIX: A square matrix is said to non-singular iff .

INVERSE OF A MATRIX

If is any square matrix then the inverse of the matrix is given by

Where.

NOTE: A square matrix is invertible iff or the square matrix should be non-singular.

RESULTS:

1.

2.

3.

4.

5.

SOLUTION OF SYSTEM OF LINEAR EQUATIONS USING MATRICES

A general system of linear equation in three variables is of the type

Where = co-efficient of ,=constant in equation ,= variable in equation and

METHOD OF SOLVING A SYSTEM OF LINEAR EQUATION IN THREE VARIABLES

Step 1. Assume

;;

Step2. The above system of linear equations can bewritten as the matrix equation as follows:

Step3. Then the solution of the variable matrix ‘’ can be found as follows:

If then the system of equations has unique solution given by

Where .

NOTE: If then there are two possibilities:

CASE 1. Ifthen the system may have infinite solution or no solution.

CASE 2. Ifthen the system has no solution.