Determinant

A determinant of order three consisting of 3 rows and 3 columns is written as:

| a1 b1 c1 |
| a2 b2 c2 |
| a3 b3 c3 |
 
And is equal to:

a1(b2c3 - b3c2) - b1(a2c3 - a3c2) + c1(a2b3 - a3b2)

= a1b2c3 - a1b3c2 - b1a2c3 + b1a3c2 + c1a2b3 - c1a3b2

Properties Of Determinants

1. If rows be changed into columns and columns into rows, the determinant remains unaltered.
2. If any two rows (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original determinant.
3. If two rows (or two columns) in a determinant have corresponding entries that are equal (or proportional), the value of determinant is equal to zero.
4. If each of the entries of one row (or columns) of a determinant is multiplied by a nonzero constant k, then the determinant gets multiplied by k.
5. If each entry in a row (or column) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or more determinants.
6. If to each element of a line (row or column) of a determinant be added the equi-multiples of the corresponding elements of one or more parallel lines, the determinant remains unaltered.
7. If each entry in any row (or any column) of determinant is zero, then the value of determinant is equal to zero.
8. If a determinant D vanishes for x = a, then (x-a) is a factor of D. In other words, if two rows (or two columns) become identical for x = a, then (x-a) is a factor of D.
9. If in a determinant (of order three or more) the elements in all the rows (columns) are in A.P. with same or different common difference, the value of the determinant is zero.
 
Also Check: Diffrential Equation

Examples 

Examples 1

The value of:

| 1 a a²-bc |
| 1 b b²-ca |
| 1 c c²-ab |

is (A) 0 (B) 1 (C) -1 (D) abc

Solution: (A) 0

Using the property that if elements in all rows are in A.P. with same or different common differences, the determinant equals zero.

 Example 2

If a ≠ p, b ≠ q, c ≠ r and:

| p b c |
| a q c | = 0
| a b r |

then the value of:

| p q r |
| p-a q-b r-c |

is (A) 1 (B) -1 (C) 0 (D) 2

Solution: (D) 2

DIFFERENTIATION OF A DETERMINANT

If Δ(x) = |a1(x) b1(x)|, then Δ'(x) = |a1'(x) b1'(x)| + |a1(x) b1(x)|

           |a2(x) b2(x)|                 |a2(x) b2(x)|   |a2'(x) b2'(x)|

For a 3×3 determinant, if Δ(x) has elements that are functions of x in one row (or column), then:

Δ'(x) = determinant with that row (or column) differentiated while others remain unchanged.

Example 3

If fr(x), gr(x), hr(x) where r = 1, 2, 3 are polynomials in x such that fr(a) = gr(a) = hr(a), r = 1, 2, 3 and:

F(x) = | f1(x) f2(x) f3(x) |
       | g1(x) g2(x) g3(x) |
       | h1(x) h2(x) h3(x) |

then F(a) = 0.

Solution: Since at x = a, all corresponding elements become equal, two rows become identical, making F(a) = 0.

SUMMATION OF DETERMINANTS

Let Δr = |fr a l|, where a, b, c, l, m, n are constants independent of r.
              |gr b m|
              |hr c n|

Then Σ(r=1 to n) Δr = |Σfr a l|
                              |Σgr b m|
                              |Σhr c n|

DETERMINANTS INVOLVING INTEGRATIONS

Let Δ(x) = |f(x) g(x) h(x)|, where a, b, c, l, m, n are constants.
              |a     b     c   |
              |l     m     n   |

Then ∫[a to b] Δ(x)dx = |∫f(x)dx ∫g(x)dx ∫h(x)dx|
                                 |a        b        c       |
                                 |l        m        n       |
Note: If elements of more than one column or row are functions of x, integration can only be done after evaluation/expansion of the determinant.

SPECIAL DETERMINANTS

1. |1 1 1 | = (a-b)(b-c)(c-a)
     |a b c |
     |a² b² c²|

2. |1 1 1 | = (a-b)(b-c)(c-a)(a+b+c)
     |a b c |
     |a³ b³ c³|

3. |1 1 1 | = (a-b)(b-c)(c-a)(ab+bc+ca)
     |a b c |
     |a² b² c²|

4. |1 1 1 | = (a-b)(b-c)(c-a)(a+b+c-ab-ac-ca)
     |a b c |
     |a⁴ b⁴ c⁴|

LINEAR EQUATIONS

Homogeneous System

The system:

a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 0

has a non-trivial solution if and only if:

| a1 b1 c1 |
| a2 b2 c2 | = 0
| a3 b3 c3 |

  • If Δ ≠ 0: only trivial solution (x = y = z = 0)

  • If Δ = 0: infinite solutions (non-trivial solutions exist)

CRAMER'S RULE

For the non-homogeneous system:

a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3

If Δ ≠ 0, then:

x = Δx/Δ, y = Δy/Δ, z = Δz/Δ

where:

Δ = | a1 b1 c1 |     Δx = | d1 b1 c1 |
    | a2 b2 c2 |          | d2 b2 c2 |
    | a3 b3 c3 |          | d3 b3 c3 |

Δy = | a1 d1 c1 |     Δz = | a1 b1 d1 |
      | a2 d2 c2 |          | a2 b2 d2 |
      | a3 d3 c3 |          | a3 b3 d3 |

System Classifications:

Consistent Independent: Δx, Δy, Δz ∈ ℝ and Δ ≠ 0 (unique solution)
Consistent Dependent: Δx = Δy = Δz = 0 and Δ = 0 (infinite solutions)
Inconsistent: At least one of Δx, Δy, Δz ≠ 0 and Δ = 0 (no solution)

SOLVED EXAMPLES

Example 1

| 109 102 95 |
| 6 13 20 | = 0
| 1 -6 -13 |
Elements in each row are in A.P., so determinant = 0.

Example 2

| a b aα+b |
| b c bα+c |
| aα+b bα+c 0 |
= 0 when a, b, c are in G.P.

Example 3

If f(x) = |cos x 1 0 |

|1 2cos x 1 |, then ∫[0 to π/2] f(x)dx = -1/3
|0 1 2cos x|

FORMULAE AND CONCEPTS AT A GLANCE

  1. The determinant remains unaltered if its rows are changed into columns and the columns into rows.
  2. If all the elements of a row (or column) are zero, then the determinant is zero.
  3. If the elements of a row (column) are proportional (or identical) to the elements of any other row (column), then the determinant is zero.
  4. The interchange of any two rows (columns) of the determinant changes its sign.
  5. If all the elements of a row (column) of a determinant are multiplied by a non-zero constant then the determinant gets multiplied by the same constant.
  6. A determinant remains unaltered under a column (Ci) operation of the form Ci + λCj + μCk (j,k ≠ i) or a row (Ri) operation of the form Ri + λRj + μRk (j,k ≠ i).
  7. If each element in any row (column) is the sum of r terms, then the determinant can be expressed as the sum of r determinants.
  8. If the determinant Δ = f(x) and f(a) = 0, then (x – a) is a factor of the determinant. In other word, if two rows (or two columns) become proportional (identical) for x = a then (x – a) is a factor of determinant. In general, if r rows become identical for x = a then (x – a)^(r-1) is a factor of the determinant.
  9. If in a determinant (of order three or more) the elements in all the rows (columns) are in A.P. with same or different common difference, the value of the determinant is zero.
  10. The determinant value of an odd order skew symmetric determinant is always zero.

SOLVED EXAMPLES

Example 1

109 102 95
6 13 20
1 -6 -13
is equal to

Options: (A) constant other than zero (B) zero (C) 100 (D) –1997

Solution: (A) We know that in a determinant, if the elements in all the rows (columns) are in AP with same or different common difference, the value of the determinant is zero.

Example 2

The determinant

a+αb b aα+b
b+βc c bβ+c
aα+b bβ+c 0
= 0 if

Options: (A) a, b, c are in A.P. (B) a, b, c are in G.P. (C) a, b, c are in H.P. (D) none of these

Solution: (B) After applying R₃ → R₃ - (R₁ + R₂), we get:
This determinant is zero if (α² + βα + c)(αc – b²) = 0
⟹ b² = αc ⟹ a, b, c are in G.P.

Example 3

If f(x) =

cos x 1 0
1 2cos x 1
0 1 2cos x

then ∫₀π/2 f(x) dx is equal to

Options: (A) 1/4 (B) –1/3 (C) 1/2 (D) 1

Solution: (B) Expanding the given determinant, f(x) = 4cos³x – 3cos x = cos 3x
∫₀π/2 f(x) dx = ∫₀π/2 cos 3x dx = [sin 3x/3]₀π/2 = -1/3

Example 4

If A, B, C are the angles of a triangle and

sin²A cotA 1
sin²B cotB 1
sin²C cotC 1

then Δ is equal to

Options: (A) constant other than zero (B) not a constant (C) 0 (D) none of these

Solution: (C) After applying R₂ → R₂ – R₁, R₃ → R₃ – R₁, the determinant becomes zero due to identical elements.

Example 5

If Δ =

x+a b c
a x+b c
a b x+c
= 0, then the non zero root is

Options: (A) a + b + c (B) abc (C) – a – b –c (D) none of these

Solution: (C) Applying C₁ → C₁ + C₂ + C₃
(x + a + b + c) times a determinant = 0 ⟹ x = –a–b–c

 

ASSIGNMENT PROBLEMS

Problem 1

The system of equations x + 2y + 3z = 4, 2x + 3y + 4z = 5, 3x + 4y + 5z = 6 has

Options: (A) infinitely many solution (B) no solution (C) unique solution (D) none of these

Problem 2

If

x+1 x+2 x+a
x+2 x+3 x+b
x+3 x+4 x+c
= 0, then a, b, c are in

Options: (A) A.P (B) G.P (C) H.P (D) none of these

Problem 3

Δ =

p 2+i i
2-i q 3+i
1-i 3-i r

(p, q, r ∈ real) is always

Options: (A) real (B) imaginary (C) zero (D) none of these

Problem 4

If a, b, c are roots of x³ + px² + q = 0, then

a b c
b c a
c a b
is equal to

Options: (A) –p (B) q (C) p³ (D) q³

Problem 5

If the determinant

a 1 1
1 b 1
1 1 c
is positive, then abc is greater than

Options: (A) 8 (B) 10 (C) 12 (D) none of these

Problem 6

If

4x 6x+2 8x+1
6x+2 9x+3 12x
8x+1 12x 16x+2
= 0, then x is equal to

Options: (A) x = 0 (B) x = -11 (C) x = 97 (D) x = –11/97

Problem 7

The value of θ lying between 0 and π/2 and satisfying:

1+cos²θ sin²θ 4sin²4θ
cos²θ 1+sin²θ 4sin²4θ
cos²θ sin²θ 1+4sin²4θ
= 0

Options: (A) (7π/24, 11π/24) (B) (8π/11, π/12) (C) (7π/12, 11π/24) (D) none

ANSWERS TO ASSIGNMENT

1. A
2. A
3. A
4. C
5. A
6. D
7. A

Frequently Asked Questions

A determinant is a scalar value calculated from a square matrix (e.g., 2×2, 3×3) that provides important information about the matrix, such as whether it is invertible.

  • Non-zero determinant → The matrix is invertible (non-singular).

  • Zero determinant → The matrix is not invertible (singular), and its rows/columns are linearly dependent.

 

  1. Row/Column Swap → Changes the sign of the determinant.

  2. Identical Rows/Columns → Determinant = 0.

  3. Scalar Multiplication → Multiplying a row/column by kkk multiplies determinant by kkk.

  4. Addition of Rows/Columns → Adding a multiple of one row to another doesn’t change the determinant.

  5. Triangular Matrix → Determinant = product of diagonal elements.

If two rows or columns are proportional (e.g., one is a multiple of the other), the determinant is zero.

Yes. The determinant can be positive, negative, or zero depending on the arrangement and orientation of the matrix’s rows/columns.

  • Finding the inverse of a matrix.

  • Solving systems of linear equations (Cramer’s Rule).

  • Calculating areas and volumes in geometry.

  • Analyzing linear transformations.

For a 2×2 matrix, the absolute value of the determinant represents the area of the parallelogram formed by its column vectors.
For a 3×3 matrix, it represents the volume of the parallelepiped.

Using Cramer’s Rule, the solution to a system of nnn equations in nnn variables can be found using determinants of the coefficient matrix and modified matrices.

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