DEFINITION OF DETERMINANTS

The determinant is the scalar value associated with a matrix. Every matrix has a unique value of the determinant. The determinant is defined only for square matrices.

For a square matrix A, the Determinant is denoted as  or det (A).

For example, |A| =  The determinant of A is 

For every square matrix, we can associate a number with it, and that will be called its determinant.

For matrix A = , Determinant can be written as |A| =  = det (A).

HOW TO CALCULATE THE DETERMINANT OF THE MATRIX?

 Method I :-

• i. To find the determinant of a matrix having order One

Let a matrix A is defined as

                                                 A = [x]

Then, the determinant of A will be x itself, i.e. |A| = x.

• ii. To Find the determinant of a matrix having order 2 × 2

Let a matrix A is defined as

 A = 

Determinant is, |A| =  = ad – bc

• iii. To find the determinant of a matrix having order 3 × 3

Let a square matrix, A = [aij ]3×3, then

                                                |A| =  a(ei – fh) -b(di – gi) + c(dh-eg)

Expansion along 1st Row:-

|A| =  a(ei – fh) -b(di – gi) + c(dh-eg)

Similarly, we can expand the matrix along any row or column in order to find its determinant.

NOTE- Sign convention will be used according to (-1)i+j (aij)).

PROPERTIES OF DETERMINANT-

1) If any row or column of a matrix is zero then its determinant will be zero.

2) If any two rows or columns have equal or proportional values then its determinant will be zero.

3) If we exchange two rows or two columns then the value of the determinant will change by a minus sign.

4) Elementary row or column transformations in a matrix do not change the value of its determinant.

5) |AT| = |A|T

6) |kA| = kn |A| ; n is the order of the matrix.

IMPORTANT RESULTS FOR DETERMINANT-

1) The determinant of a skew-symmetric matrix having odd order is zero.

2) For a diagonal matrix, its determinant is equal to its diagonal elements.

Multiplication in Determinant By a Scalar

If a number is multiplied by a determinant, then the number will get multiplied into any one row or any one column.

SINGULAR MATRIX-

If a matrix has its determinant equal to zero, then the matrix will be called SINGULAR MATRIX.

NON- SINGULAR MATRIX-

If a matrix has a non-zero determinant, then the matrix will be NON-SINGULAR.

For the system of equations to have ‘a unique solution’, the matrix must be non-singular.

MINORS OF AN ELEMENT-

Minor of an element aij will be obtained by removing ith row and jth column.

 Minor of an element is denoted as Mij.

For example,

• For a 3 × 3 determinant ,  

Let say A =

Minor of given Determinants are:

• M11 =     

•  M12 =  

• M13 =

• M21  =         

• M22 =    

• M23 =

COFACTORS-

For an element aij, Cofactor is denoted by Cij and it will obtained by,

Cij = (-1)(i+j).Mij

Now we can easily understand that how we were finding determinant,

|A| = a11C11 + a12C12 + a13C13   (Expansion along 1st row)

LAPLACE FORMULA-

By using this formula, We can express the determinant of the matrix in terms of a linear combination of different smaller matrices (that are called minors).

|A| = i+j aij.Mij

n = order of matrix

Mij = Minor of matrix

(-1)i+j.Mij = Cofactor Of matrix

ADJOINT OF A MATRIX-

If we form a matrix of Cofactors of the matrix and if we take its transpose, then the Adjoint Matrix will form.

Step to find the Adjoints of the matrix :

• Step 1: Determine the Cofactor for each element in the matrices.
• Step 2: Using the cofactors, create a new matrix and extend the cofactors, resulting in a matrix.
• Step 3: Now find the matrix’s transpose, which you got from Step 2.

PROPERTIES OF ADJOINT MATRIX-

1. A(adj A) = |A|In = (adj A)A
2. |adj A| = |A|n-1
3. Adj (adj A) = |A|n-2 A
4. |Adj (adj A)| = |A|(n-1)^2
5. adj (AT) = (adj A)T
6. Adj (AB) = (adj B) (adj A)
7. Adj (Am) = (adj A)m ,(m ∈ N)
8. Adj (kA) = kn-1 (adj A) ,(k ∈ R)

INVERSE OF A MATRIX-

For every non-singular square matrix of order n × n, the inverse of the matrix will exist, if it follows the Property,

A.A-1 = I = A-1.A

Where, A-1 is the inverse of a matrix

It can be calculated by,

A-1 =

     PROPERTIES OF INVERSE MATRIX-

1. Every invertible matrix will have a unique inverse.
2. (AB)-1 = B-1A-1, if A and B both are an invertible matrix of the same order and AB is also invertible.
3. If A is an invertible square matrix, so its Transpose will also be invertible and follows the property (AT)-1 = (A-1)T
4. |A-1| = |A|-1

APPLICATION OF DETERMINANT-

1. To find Area Of the Triangle

Let, you have to find an area of a triangle with vertices (x1, y1) (x2, y2) (x3, y3)

          Area =  ½ |A|

and |A| is equal to 

 If the determinant value comes negative, the neglect negative sign as Area cannot be negative,

2. To find solution Of Linear Equations

Consistent System : System Of Equations whose solution exists (one or can be more than one).

Inconsistent System: System Of Equations whose solution does not exist.

System Of Linear Simultaneous Equations

Let us consider a system,

A =     

X =             

B =

For a non-singular matrix A,  inverse for this matrix will exist, then

AX = B

X = A-1.B

Case 1 : |A| ≠ 0, inverse exists.

⇒ AX = B

⇒ X = A-1.B

Then, it will give a unique solution.

 Case 2 : |A| = 0, inverse does not exist.

We have AX = B

                              ⇒ ((adj A) A) X = (adj A) B

                              ⇒ |A| X = (adj A) B

1. If, (adj A) B = 0, the system has infinitely many solutions.
2. If, (adj A) B ≠ 0, the system has no  solutions.

Example,

             Solve the system of equations

2x + 5y = 1

3x + 2y = 7

Sol. – A =      X =        B =      

⇒ |A| = -11, Hence A is an invertible matrix and have a unique solution.

A-1 = –  

X = A–1B = –       

 = –   =

So, x = 3 and y = -1

CONCLUSION-

 The determinant is a very important part of completing Matrices theory. And, it has so many uses also like the area of a triangle, solutions of equations. Above we have discussed the cofactors on determinants, adjoint of determinants, the properties to be kept in mind while solving determinants. There are multiple applications of determinants that are used to do mathematical calculations.