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4.1 Overview
To every square matrix A = [ aij ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where aij is the ( i , j ) th element of A.
If A^
a b c d
=^ ^
, then determinant of A, denoted by |A| (or det A), is given by
|A|
a b c d = (^) = ad – bc.
Remarks
(i) Only square matrices have determinants.
(ii) For a matrix A, A is read as determinant of A and not, as modulus of A.
4.1.1 Determinant of a matrix of order one
Let A = [ a ] be the matrix of order 1, then determinant of A is defined to be equal to a.
4.1.2 Determinant of a matrix of order two
Let A = [ aij ] =
a b c d
be a matrix of order 2. Then the determinant of A is defined
as: det (A) = |A| = ad – bc.
4.1.3 Determinant of a matrix of order three
The determinant of a matrix of order three can be determined by expressing it in terms of second order determinants which is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows (R 1 , R 2 and R 3 ) and three columns (C 1 , C 2 and C 3 ) and each way gives the same value.
Chapter 4
DETERMINANTS
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Consider the determinant of a square matrix A = [ aij ]3×3, i.e.,
11 12 13 21 22 23 31 32 33
A
a a a a a a a a a
Expanding |A| along C 1 , we get
|A| = a 11 (–1)1+^ 22 23 32 33
a a a a +^ a^21 (–1)
2+1 12 13 32 33
a a a a +^ a^31 (–1)
3+1 12 13 22 23
a a a a = a 11 ( a 22 a 33 – a 23 a 32 ) – a 21 ( a 12 a 33 – a 13 a 32 ) + a 31 ( a 12 a 23 – a 13 a 22 )
Remark In general, if A = k B, where A and B are square matrices of order n , then |A| = kn^ |B|, n = 1, 2, 3.
4.1.4 Properties of Determinants
For any square matrix A, |A| satisfies the following properties.
(i) |A′| = |A|, where A′ = transpose of matrix A.
(ii) If we interchange any two rows (or columns), then sign of the determinant changes.
(iii) If any two rows or any two columns in a determinant are identical (or proportional), then the value of the determinant is zero.
(iv) Multiplying a determinant by k means multiplying the elements of only one row (or one column) by k.
(v) If we multiply each element of a row (or a column) of a determinant by constant k , then value of the determinant is multiplied by k.
(vi) If elements of a row (or a column) in a determinant can be expressed as the sum of two or more elements, then the given determinant can be expressed as the sum of two or more determinants.
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[ aij ] n × n , where A ij is the co-factor of the element aij. It is denoted by adj A.
If
11 12 13 21 22 23 31 32 33
A ,
a a a a a a a a a
= then adj
11 21 31 12 22 32 13 23 33
A A A
A A A A ,
A A A
= where A ij is co-factor of aij.
(ii) A ( adj A) = ( adj A) A = |A| I, where A is square matrix of order n.
(iii) A square matrix A is said to be singular or non-singular according as |A| = 0 or |A| ≠ 0, respectively. (iv) If A is a square matrix of order n , then | adj A| = |A| n –1. (v) If A and B are non-singular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.
(vi) The determinant of the product of matrices is equal to product of their respective determinants, that is, |AB| = |A| |B|.
(vii) If AB = BA = I, where A and B are square matrices, then B is called inverse of A and is written as B = A–1. Also B–1^ = (A–1)–1^ = A.
(viii) A square matrix A is invertible if and only if A is non-singular matrix.
(ix) If A is an invertible matrix, then A–1^ =
| A | ( adj^ A)
4.1.8 System of linear equations
(i) Consider the equations: a 1 x + b 1 y + c 1 z = d 1 a 2 x + b 2 y + c 2 z = d 2 a 3 x + b 3 y + c 3 z = d 3 , In matrix form, these equations can be written as A X = B, where
A =
1 1 1 1 2 2 2 2 3 3 3 3
, X and B
a b c x d a b c y d a b c z d
(ii) Unique solution of equation AX = B is given by X = A–1B, where |A| ≠ 0.
DETERMINANTS 69
(iii) A system of equations is consistent or inconsistent according as its solution exists or not. (iv) For a square matrix A in matrix equation AX = B (a) If |A| ≠ 0, then there exists unique solution. (b) If |A| = 0 and ( adj A) B ≠ 0, then there exists no solution. (c) If |A| = 0 and ( adj A) B = 0, then system may or may not be consistent.
4.2 Solved Examples
Short Answer (S.A.)
Example 1 If
x x
= (^) , then find x.
Solution We have
x x = (^). This gives
2 x^2 – 40 = 18 – 40 ⇒ x^2 = 9 ⇒ x = ± 3.
Example 2 If
2 2 1 2
x x y y yz zx xy z z x^ y^ z
∆ = ∆ = (^) , then prove that ∆ + ∆ 1 = 0.
Solution We have (^1)
yz zx xy x y z
Interchanging rows and columns, we get
1
yz x zx y xy z
2 2 2
x xyz x y xyz y xyz z xyz z
DETERMINANTS 71
p x q x p x q q x
= − − Applying R 1 → R 1 + R 2
Expanding along C 1 , we have
∆ = ( x − p )( px + x^2^ − 2 q^2 )=^ ( x − p )( x^2 + px − 2 q^2 )
Example 5 If
b a c a a b c b a c b c
, then show that ∆ is equal to zero.
Solution Interchanging rows and columns, we get
a b a c b a b c c a c b
Taking ‘–1’ common from R 1 , R 2 and R 3 , we get
3
b a c a a b c b a c b c
⇒ (^2) ∆ = 0 or (^) ∆ = 0
Example 6 Prove that (A–1)′^ = (A′)–1, where A is an invertible matrix.
Solution Since A is an invertible matrix, so it is non-singular.
We know that |A| = |A′|. But |A| ≠ 0. So |A′| ≠ 0 i.e. A′ is invertible matrix.
Now we know that AA–1^ = A–1^ A = I.
Taking transpose on both sides, we get (A–1)′ A′ = A′ (A–1)′ = (I)′ = I
Hence (A–1)′ is inverse of A′, i.e., (A′)–1^ = (A–1)′
Long Answer (L.A.)
Example 7 If x = – 4 is a root of
x x x
∆ = (^) = 0, then find the other two roots.
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Solution Applying R 1 → (R 1 + R 2 + R 3 ), we get
x x x x x
Taking ( x + 4) common from R 1 , we get
1 1 1 ( 4) 1 1 3 2
x x x
Applying C 2 → C 2 – C 1 , C 3 → C 3 – C 1 , we get
x x x
Expanding along R 1 ,
∆ = ( x + 4) [( x – 1) ( x – 3) – 0]. Thus, ∆ = 0 implies
x = – 4, 1, 3
Example 8 In a triangle ABC, if
2 2 2
1 sin A 1 sin B 1 sin C 0 sinA +sin A sinB+sin B sinC+sin C
then prove that ∆ABC is an isoceles triangle.
Solution Let ∆ = 2 2 2
1 sin A 1 sin B 1 sin C sinA +sin A sinB+sin B sinC+sin C
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or sinθ = 0 or (2sinθ – 1) = 0 or (2sinθ + 3) = 0
or sinθ = 0 or sinθ =
2 (Why ?).
Objective Type Questions
Choose the correct answer from the given four options in each of the Example 10 and 11.
Example 10 Let
2 2 1 2
A 1 A B C
B 1 and C 1
x x y y x y z z z zy^ zx^ xy
∆ = ∆ = , then
(A) ∆ 1 = – ∆ (B) ∆ ≠ ∆ 1
(C) ∆ – ∆ 1 = 0 (D) None of these
Solution (C) is the correct answer since (^1)
A B C
x y z zy zx xy
A
B
C
x yz y zx z xy
2 2 2
A
1 B
C
x x xyz y y xyz xyz z z xyz
2 2 2
A 1
B 1
Cz 1
x x xyz y y xyz z
Example 11 If x , y ∈ R , then the determinant
cos sin 1 sin cos 1 cos( ) sin( ) 0
x x x x x y x y
lies
in the interval
(A) ^ − 2, 2 (B) [–1, 1]
(C) ^ − 2,1 (D) ^ −1,^ − 2,
Solution The correct choice is A. Indeed applying R 3 → R 3 – cos y R 1 + sin y R 2 , we get
DETERMINANTS 75
cos sin 1 sin cos 1 0 0 sin cos
x x x x y y
Expanding along R 3 , we have
∆ = (sin y – cos y ) (cos^2 x + sin^2 x )
= (sin y – cos y ) = 2 1 sin 1 cos 2 2
(^) y − y
= 2 cos^4 sin^^ y^ sin^4 cos y π^ − π =^2 sin ( y^ –^4
π )
Hence – 2 ≤ ∆ ≤ 2.
Fill in the blanks in each of the Examples 12 to 14.
Example 12 If A, B, C are the angles of a triangle, then
2 2 2
sin A cotA 1 sin B cotB 1 ................ sin C cotC 1
Solution Answer is 0. Apply R 2 → R 2 – R 1 , R 3 → R 3 – R 1.
Example 13 The determinant
is equal to ...............
Solution Answer is 0.Taking 5 common from C 2 and C 3 and applying
C 1 → C 3 – 3 C 2 , we get the desired result.
Example 14 The value of the determinant
DETERMINANTS 77
Example 18 If –
A 1 2 , A 1 3 3
x
y
= ^ ^ = ^ − −
then x = 1, y = – 1.
Solution True
4.3 EXERCISE
Short Answer (S.A.)
Using the properties of determinants in Exercises 1 to 6, evaluate:
x x x x x
a x y z x a y z x y a z
2 2 2 2 2 2
xy xz x y yz x z zy
x x y x z x y y z y x z y z z
x x x x x x x x x
a b c a a b b c a b c c c a b
Using the proprties of determinants in Exercises 7 to 9, prove that:
2 2 2 2 2 2
y z yz y z z x zx z x x y xy x y
8.^4
y z z y z z x x xyz y x x y
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2 3
a a a a a a
10. If A + B + C = 0, then prove that
1 cos C cos B cos C 1 cos A 0 cos B cos A 1
11. If the co-ordinates of the vertices of an equilateral triangle with sides of length
‘ a ’ are ( x 1 , y 1 ), ( x 2 , y 2 ), ( x 3 , y 3 ), then
2 (^1 1 ) 2 2 3 3
x y x y a x y
12. Find the value of θ satisfying
1 1 sin 3 4 3 cos 2 0 7 7 2
θ (^) − θ = (^) − −
13. If
x x x x x x x x x
−^ +^ +
, then find values of x.
14. If a 1 , a 2 , a 3 , ..., ar are in G.P., then prove that the determinant
1 5 9 7 11 15 11 17 21
r r r r r r r r r
a a a a a a a a a
is independent of r.
15. Show that the points ( a + 5, a – 4), ( a – 2, a + 3) and ( a , a ) do not lie on a straight line for any value of a. 16. Show that the ∆ABC is an isosceles triangle if the determinant
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23. If x + y + z = 0, prove that
xa yb zc a b c yc za xb xyz c a b zb xc ya b c a
Objective Type Questions (M.C.Q.)
Choose the correct answer from given four options in each of the Exercises from 24 to 37.
24. If
x x
= (^) , then value of x is
(A) 3 (B) ± 3
(C) ± 6 (D) 6
25. The value of determinant
a b b c a
b a c a b
c a a b c
(A) a^3 + b^3 + c^3 (B) 3 bc (C) a^3 + b^3 + c^3 – 3 abc (D) none of these
26. The area of a triangle with vertices (–3, 0), (3, 0) and (0, k ) is 9 sq. units. The value of k will be (A) 9 (B) 3 (C) – 9 (D) 6 27. The determinant
2 2 2 2
b ab b c bc ac ab a a b b ab bc ac c a ab a
equals
(A) abc ( b – c ) ( c – a ) ( a – b ) (B) ( b – c ) ( c – a ) ( a – b ) (C) ( a + b + c ) ( b – c ) ( c – a ) ( a – b ) (D) None of these
DETERMINANTS 81
28. The number of distinct real roots of
sin cos cos cos sin cos 0 cos cos sin
x x x x x x x x x
= (^) in the interval
− π^ ≤ x ≤ πis
(A) 0 (B) 2 (C) 1 (D) 3
29. If A, B and C are angles of a triangle, then the determinant
1 cos C cos B cos C 1 cos A cos B cos A 1
is equal to
(A) 0 (B) – 1
(C) 1 (D) None of these
30. Let f ( t ) =
cos 1 2sin 2 sin
t t t t t t t t
, then (^02) lim^ ( ) t
f t → (^) t is equal to
(A) 0 (B) – 1
(C) 2 (D) 3
31. The maximum value of
1 1 sin 1 1 cos 1 1
∆ = + θ
is (θ is real number)
(A)
2 (B)^
(C) 2 (D)
DETERMINANTS 83
37. There are two values of a which makes determinant, ∆ =
a a
− = 86, then
sum of these number is (A) 4 (B) 5 (C) – 4 (D) 9
Fill in the blanks
38. If A is a matrix of order 3 × 3, then |3A| = _______. 39. If A is invertible matrix of order 3 × 3, then |A–1^ | _______. 40. If x , y , z ∈ R, then the value of determinant
( ) ( )
( ) ( )
( ) ( )
x x x x x x x x
x x x x
is
equal to _______.
41. If cos2θ = 0, then
0 cos sin^2 cos sin 0 _________. sin 0 cos
θ θ θ θ = θ θ
42. If A is a matrix of order 3 × 3, then (A^2 )–1^ = ________. 43. If A is a matrix of order 3 × 3, then number of minors in determinant of A are ________. 44. The sum of the products of elements of any row with the co-factors of corresponding elements is equal to _________. 45. If x = – 9 is a root of
x x x
= 0, then other two roots are __________.
xyz x z y x y z z x z y
= __________.
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47. If f ( x ) =
17 19 23 23 29 34 41 43 47
x x x x x x x x x
= A + B x + C x^2 + ..., then
A = ________.
State True or False for the statements of the following Exercises:
48. (^) ( ) 3 – A = (^) ( ) 13 A−^ , where A is a square matrix and |A| ≠ 0. 49. ( a A)–1 =^
1 A–
a , where^ a^ is any real number and A is a square matrix.
50. |A–1| ≠ |A|–1^ , where A is non-singular matrix. 51. If A and B are matrices of order 3 and |A| = 5, |B| = 3, then |3AB| = 27 × 5 × 3 = 405. 52. If the value of a third order determinant is 12, then the value of the determinant formed by replacing each element by its co-factor will be 144.
x x x a x x x b x x x c
, where a , b , c are in A.P..
54. | adj. A| = |A|^2 , where A is a square matrix of order two. 55. The determinant
sin A cos A sin A + cos B sin B cos A sin B+ cos B sin C cos A sin C + cos B
is equal to zero.
56. If the determinant +
x a p u l f y b q v m g z c r w n h
splits into exactly K determinants of
order 3, each element of which contains only one term, then the value of K is 8.
