Short Notes and Video Lecture: Determinants

Determinants

determinant is a number that can be associated to every square matrix A = [aij] of order n. It can be thought of as a function that associates each square matrix with a unique number (real or complex). We denote the determinant by |A| or det A or Δ.

Only square matrices have determinants.

 

Determinant of a matrix of order one

Let  A = [a] be a matrix of order 1, then |A| = a.

Determinant of a matrix of order two

Determinant of a matrix of order three

A determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column). This can be done in six ways: corresponding to each row and each column. For example:

Expansion along R1

Step 1: Multiply first element a11 of R1 by (-1)sum of suffixes in a11 i.e. (-1)(1+1) and then multiply this product with the second order determinant obtained by deleting the elements of the row and the column in which the concerned element (a11) lies i.e. R1 and C1.

Step 2: Multiply second element a12 of R1 by (-1)sum of suffixes in a12 i.e. (-1)(1+2) and then multiply this product with the second order determinant obtained by deleting the elements of the row and the column in which the concerned element (a12) lies i.e. R1 and C2.

Step 3: Similarly, repeat the same process for the third element a13 of R1.

 

Step 4: Now the expansion of determinant of A i.e |A| is the sum of all three terms obtained in steps 1, 2 and 3 above.

  • Expanding a determinant along any row or column gives the same value.
  • For easier calculations, we expand the determinant along that row or column which contains maximum number of zeros.
  • If A = kB where A and B are square matrices of order n, then |A| = kn|A|, where n = 1, 2,3.

Properties of Determinants

Property 1: 

The value of the determinant remains unchanged if its rows and columns are interchanged, i.e. |A| = |A’|.

Property 2: 

If any two rows (or columns) of a determinant are interchanged, then sign of the determinant changes.

Property 3: 

If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of the determinant is zero.

Property 4: 

If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

By this property, we can take out any common factor from any one row or any one column of a given determinant.

If corresponding elements of any two rows (or columns) of a determinant are proportional (in the same ratio), then its value is zero.

Property 5: 

If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

Property 6: 

If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of the determinant remains the same.

Hence, Δ=Δ1

Area of a Triangle

We know, the area of a triangle whose vertices are (x1,y1), (x2,y2) and (x3,y3), is given by:

Now, we can write this expression in the form of a determinant:

Since area is a positive quantity, we always take the absolute value of the determinant.

If area is given, use both positive and negative values of the determinant for calculation.

The area of the triangle formed by three collinear points is zero.

Minors and Cofactors

Minors

Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element aij is denoted by Mij.

Minor of an element of a determinant of order n (n ≥ 2) is a determinant of order n-1.

Cofactors

Cofactor of an element aij, denoted by Aij is defined by Aij = (-1)i+j Mij, where Mij is minor of aij.

  • If elements of a row (or column) are multiplied with their corresponding cofactors, then their sum is equal to the value of the determinant of A, i.e., |A|.
  • If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero.

Adjoint and Inverse of a Matrix

The Adjoint of a square matrix A= [aij] is defined as the transpose of the matrix [Aij], where Aij is the cofactor of the element aij. Adjoint of a matrix A is denoted by adj A.

Also, for a square matrix of order 2, given bythe adj A can also be obtained by interchanging a11 and a22 and by changing the signs of a12 and a21, i.e.,

If A be any given matrix of order n, then A(adj A) = (adj A) = |A|I, where I is the identity matrix of order n.

Singular and Non-singular Matrices

A square matrix A is said to be singular if |A| = 0.

And, if |A| ≠ 0, we call it a non-singular matrix.

If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.

|AB| = |A||B|, where A and B are square matrices of the same order.

If A is a square matrix of order n, then |adj (A)| = |A|n-1 .

Invertible Matrices

A square matrix A is invertible if and only if A is non-singular matrix.

Applications of Determinants and Matrices: System of Linear Equations

  • A system of equations is said to be consistent if its solution (one or more) exists.
  • A system of equations is said to be inconsistent if its solution does not exist.

Case 1: If A is a non-singular matrix, then its inverse exists.

Now, AX = B ⇒ X = A-1 B

This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method.

Case 2: If A is a singular matrix, then |A| = 0.
In this case, we calculate (adj A)B.

If (adj A)B ≠ O, (O being zero matrix), then solution does not exist and the system of equations is called inconsistent.
If (adj A)B = O, then the system may be either consistent or inconsistent, i.e., it may either have infinitely many solutions or no solution.

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