Consider the system of linear equations:

l₁x + m₁y = p l₂x + m₂y = q

If l₁/l₂ ≠ m₁/m₂  the system of equations can have a unique solution.

If l₁/l₂ = m₁/m_{2 ≠} p/q the system of equations can have a no solution.

If l₁/l₂ = m₁/m_{2 =} p/q the system of equations can have infinitely many solution.

A system of linear equations can be expressed in the form of a matrix.

The given system can be represented in matrix form as: $\left[\begin{array}{cc}{\text{l}}_{\text{1}}& {\text{m}}_{\text{1}}\\ {\text{l}}_{\text{2}}& {\text{m}}_{\text{2}}\end{array}\right]$ $\left[\begin{array}{c}\text{x}\\ \text{y}\end{array}\right]$  = $\left[\begin{array}{c}\text{p}\\ \text{q}\end{array}\right]$

Consider the 2x2 matrix, $\left[\begin{array}{cc}{\text{l}}_{\text{1}}& {\text{m}}_{\text{1}}\\ {\text{l}}_{\text{2}}& {\text{m}}_{\text{2}}\end{array}\right]$

The real number, l₁m₂ - l₂m₁, which is associated with the matrix $\left[\begin{array}{cc}{\text{l}}_{\text{1}}& {\text{m}}_{\text{1}}\\ {\text{l}}_{\text{2}}& {\text{m}}_{\text{2}}\end{array}\right]$, can be obtained by

multiplying the diagonal elements and then subtracting the products.

Determinant of matrix $\left[\begin{array}{cc}{\text{l}}_{\text{1}}& {\text{m}}_{\text{1}}\\ {\text{l}}_{\text{2}}& {\text{m}}_{\text{2}}\end{array}\right]$  = l₁m₂ – l₂m₁

Let A =Set of square matrices B =Set of numbers (real and complex)

Let a function, f: A → B, be defined by f(X) = b such that X ∈ A and b ∈ B. Then f(X) is called the determinant of X.

It is denoted by |A|, det(A) or Δ, and is read as determinant of A.

If A =  $\left[\begin{array}{cc}{\text{a}}_{\text{1}}& {\text{b}}_{\text{1}}\\ {\text{a}}_{\text{2}}& {\text{b}}_{\text{2}}\end{array}\right]$  , then |A| = det A =$\left|\begin{array}{cc}{\text{a}}_{\text{1}}& {\text{b}}_{\text{1}}\\ {\text{a}}_{\text{2}}& {\text{b}}_{\text{2}}\end{array}\right|$.

Note:

Modulus exists only for real numbers not for matrices. Hence, |A| is read as determinant of A, but not as modulus of A.

Determinants exist only for square matrices.

Determinant of a matrix of order one

Let A = [a₁₁]

Determinant of matrix A = |a₁₁| = a₁₁

Determinant of a matrix of order two

Let  A = $\left[\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\end{array}\right]$ 

Determinant of matrix A = $\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\end{array}\right|$ = a₁₁a₂₂ - a₁₂a₂₁

Determinant of a matrix of order three

Consider the general form of a 3x3 matrix, $\left[\begin{array}{ccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right]$

The determinant of a third order matrix can be determined by expanding it along a row or a column in terms of a second order determinant.This is known as the expansion of a determinant.

The determinant of matrix A can be calculated in 6 ways. They are:

(i) Expanding the determinant along R₁ (1^st row)

(ii) Expanding the determinant along R₂ (2^nd row)

(iii) Expanding the determinant along R₃ (3^rd row)

(iv) Expanding the determinant along C₁ (1^st column)

(v) Expanding the determinant along C₂ (2^nd column)

(vi) Expanding the determinant along C₃ (3^rd column)

Expanding the determinant along R₁ (1^st row):

Step1: Multiply the first element, i.e. a₁₁, by (-1)¹⁺¹, and then by the second order determinant obtained by deleting the elements of the first row and the first column of matrix A.

The product corresponding is ${\text{(-1)}}^{\text{1+1}}{\text{a}}_{\text{11}}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ .

Step2: Multiply the second element, a₁₂, by (-1)¹⁺² and then by the second order determinant obtained by deleting the entries of the first row and the second column.

The product is  ${\text{(-1)}}^{\text{1+2}}{\text{a}}_{\text{12}}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ .

Step3: Multiply the third element, a₁₃, by (-1)¹⁺³ and then by the second order determinant obtained by deleting the entries of the first row and the third column.

The product is  ${\text{(-1)}}^{\text{1+3}}{\text{a}}_{\text{13}}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$

Step 4: The sum of the results of these three steps is the required determinant of A.

|A| = ${\text{(-1)}}^{\text{1+1}}{\text{a}}_{\text{11}}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+2}}{\text{a}}_{\text{12}}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+3}}{\text{a}}_{\text{13}}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$

= a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)

= a₁₁a₂₂a₃₃ - a₁₁a₂₃a₃₂ - a₁₂a₂₁a₃₃ + a₁₂a₂₃a₃₁ + a₁₃a₂₁a₃₂ - a₁₃a₂₂a₃₁

Expanding the determinant along R₂ ( 2^nd row):

Expanding along R₂, we get

|A| =  ${\text{(-1)}}^{\text{2+1}}{\text{a}}_{\text{21}}\left|\begin{array}{cc}{\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{2+2}}{\text{a}}_{\text{22}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{2+3}}{\text{a}}_{\text{23}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$

|A| = $\text{-}{\text{a}}_{\text{21}}\left|\begin{array}{cc}{\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|\text{+}{\text{a}}_{\text{22}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|\text{-}{\text{a}}_{\text{23}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$

= -a₂₁(a₁₂a₃₃ - a₁₃a₃₂) + a₂₂(a₁₁a₃₃ - a₁₃a₃₁) - a₂₃(a₁₁a₃₂ - a₁₂a₃₁)

= -a₂₁a₁₂a₃₃ + a₂₁a₁₃a₃₂ + a₂₂a₁₁a₃₃ - a₂₂a₁₃a₃₁ - a₂₃a₁₁a₃₂ + a₂₃a₁₂a₃₁)

Expanding the determinant along R₃ ( 3^rd row):

Expanding along R₃, we get

|A| =  ${\text{(-1)}}^{\text{3+1}}{\text{a}}_{\text{31}}\left|\begin{array}{cc}{\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{3+2}}{\text{a}}_{\text{32}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{3+3}}{\text{a}}_{\text{33}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\end{array}\right|$

|A| = ${\text{a}}_{\text{31}}\left|\begin{array}{cc}{\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\end{array}\right|\text{–}{\text{a}}_{\text{32}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|\text{+}{\text{a}}_{\text{23}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\end{array}\right|$

= a₃₁(a₁₂a₂₃ - a₁₃a₂₂) - a₃₂(a₁₁a₂₃ - a₁₃a₂₁) - a₃₃(a₁₁a₂₂ - a₁₂a₂₁₎

|A| = a₁₁a₂₂a₃₃ - a₁₁a₂₃a₃₂ - a₁₂a₂₁a₃₃ + a₁₂a₂₃a₃₁ + a₁₃a₂₁a₃₂ - a₁₃a₂₂a₃₁

Expanding the determinant along C₁ ( 1^st column):

Expanding along C₁, we get

|A| =  ${\text{(-1)}}^{\text{1+1}}{\text{a}}_{\text{11}}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{2+1}}{\text{a}}_{\text{21}}\left|\begin{array}{cc}{\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{32}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{3+1}}{\text{a}}_{\text{31}}\left|\begin{array}{cc}{\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\end{array}\right|$

|A| = ${\text{a}}_{\text{11}}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{32}}\end{array}\right|\text{–}{\text{a}}_{\text{21}}\left|\begin{array}{cc}{\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|\text{+}{\text{a}}_{\text{31}}\left|\begin{array}{cc}{\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\end{array}\right|$

= a₁₁(a₂₂a₃₃ - a2₃a₃₂) - a₂₁(a₁₂a₃₃ - a₁₃a₃₂) + a₃₁(a₁₂a₂₃ - a₁₃a₂₂)

|A| = a₁₁a₂₂a₃₃ - a₁₁a2₃a₃₂ - a₂₁a₁₂a₃₃ + a₂₁a₁₃a₃₂ + a₃₁a₁₂a₂₃ - a₃₁a₁₃a₂₂

Similarly the determinant of A will be the same along columns C ₂ and C₃.

Some important points:

(i) If any row (or column) contains maximum number of zero(s), then finding the determinant along that row (or column) makes the calculation easier.

(ii) If all the elements in a row (or a column) are zeros, then the determinant of that matrix is zero.

(iii) If M and N are two square matrices such that M = kN, where k is a real number, then |M| = kⁿ|N|, where n is the order of matrices M and N.

(iv) At the time of expansion, a_ij can be multiplied by +1 or -1, depending upon whether (i + j) is even or odd, instead of multiplying it by (-1)^{i + j}.