Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2

You can Download Samacheer Kalvi 11th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2

Question 1.
Without expanding the determinant, prove thatSamacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 1
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 2

Question 2.
Show that Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 3
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 4
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 5

Question 3.
Prove that Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 6
Solution:
LHS
Taking a from C1, b from C2 and c from C3 we get
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 7
Expanding along R1 we get
(2c) (abc) (1) [ab + ab] = abc (2c) (2ab)
1 = (abc) (4abc) = 4a2b2c2
= RHS

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2

Question 4.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 8
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 9

Question 5.
Prove that Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 10
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 11

Question 6.
Show that Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 12
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 13
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 14

Question 7.
Write the general form of a 3 × 3 skew-symmetric matrix and prove that its determinant is 0.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 15

Question 8.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 16
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 17
we get – (aα2 + 2bα + c) [ac – b2]
So Δ = 0 ⇒ (aα2 + 2bα + c) (ac -b2) = – 0 = 0
⇒ aα2 + 2bα + c = 0 or ac – b2 = 0
(i.e.) a is a root of ax2 + 2bx + c = 0
or ac = b2
⇒ a, b, c are in G.P.

Question 9.
Prove that Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 18
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 19
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 20

Question 10.
If a, b, c are pth, qth and rth terms of an A.P., find the value of Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 21
Solution:
We are given a = tp,b = tq and c = tr
Let a be the first term and d be the common difference
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 22

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2

Question 11.
Show that Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 23 is divisible by x4
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 24
Multiplying R1 by a, R2 by b and R3 by c and
taking out a from C1 b from C2 and c from C3 we get
=Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 25=

Question 12.
If a, b, c are all positive, and are pth, qth and rth terms of a G.P., show that
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 26
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 27
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 28

Question 13.
Find the value of Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 29 if x, y, z ≠ 1.
Solution:
Expanding the determinant along R1
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 30

Question 14.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 31
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 32
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 33

Question 15.
Without expanding, evaluate the following determinants:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 34
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 35

Question 16.
If A is a square matrix and |A| = 2, find the value of |AAT|.
Solution:
Given |A| = 2
[Property 1: The determinant of a matrix remains unaltered if its rows are changed into columns and columns. That is, |A| = |AT|]

|AT| = |A| = 2
∴ |A AT| = |A| |AT|
= 2 × 2 = 4

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2

Question 17.
If A and B are square matrices of order 3 such that |A| = -1 and |B| = 3, find the value of |3AB|.
Solution:
Given A and B are square matrices of order 3 such that |A| = -1 and |B| = 3
[It A is a square matrix of order n then det ( kA) = |kA| = kn |A|.]
A and B are square matrices of order 3. Therefore,
AB is also a square matrix of order 3.
|3 AB| = 33 |AB|
= 27 |A| |B|
= 27 × – 1 × 3
|3 AB| = – 81

Question 18.
If λ = -2, determine the value of Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 36
Solution:
Given λ = -2
∴ 2λ = -4; λ2 = (-2)2; 3λ2 + 1 = 3 (4) + 1 = 13
6λ – 1 = 6(-2) – 1 = -13
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 37
expanding along R1
0(0) + 4 (0 + 13) + 1 (-52 + 0) = 52 – 52 = 0
Aliter: The determinant value of a skew-symmetric matrix is zero

Question 19.
Determine the roots of the equation Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 38
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 39

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2

Given the determinant value is 0
⇒ 30(1 + x) (2 – x) = 0
⇒ 1 + x = 0 or 2 – x = 0
⇒ x = -1 or x = 2
So, x = -1 or 2.

Question 20.
Verify that det (AB) = (det A) (det B) for Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 40
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 41
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 42
{(-20)(52) (-19) + (10)(38)(—49) + (2)(64)(-17)} – {(-49)(52) (2) + (-17)(38)(-20) + (-19)(64)(10)}
= (19760 – 18620 – 2176) – (-5096 + 12920 – 12160)
= (19760 + 5096 + 12160) – (18620 + 2176 + 12920)
= 37016 – 33716 = 3300 ….(3)
Now (1) × (2) = (3)
(i.e.,) (-33) (-100) = 3300
⇒ det (AB) = (det A), (det B)

Question 21.
Using cofactors of elements of the second row, evaluate |A|, where Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 43
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 44

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 Additional Problems

Question 1.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 45
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 46
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 47

Question 2.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 48
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 49

Question 3.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 50
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 51
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 52

Question 4.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 53
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 54
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 55
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 56

Question 5.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 57
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 58

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2

Question 6.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 59
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 60
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 61

Question 7.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 62
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 63

Question 8.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 64
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 65
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 66

Question 9.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 67
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 68
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.2 69

Leave a Comment

Your email address will not be published. Required fields are marked *