Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1

   

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 7.1

11th Maths Matrices And Determinants Solutions Question 1.
Construct an m × n matrix A = [aij], where aij is given by
11th Maths Matrices And Determinants Solutions Samacheer Kalvi
Solution:
(i) aij = \(\frac{(i-2 j)^{2}}{2}\)
Here m = 2, n = 3
So we have to construct a matrix of order 2 × 3
11th Maths Exercise 7.1 Answers Samacheer Kalvi

(ii) Here m = 3 and n = 4
So we have to construct a matrix order 3 × 4
The general form of a matrix of order 3 × 4 will be
11th Maths 7.1 Exercise Samacheer Kalvi
11th Maths Exercise 7.1 Samacheer Kalvi

11th Maths Exercise 7.1 Answers Question 2.
Find the values of p, q, r and s if
11th Maths Matrix Solutions Samacheer Kalvi
Solution:
When two matrices (of same order) are equal then their corresponding entries are equal.
11th Maths Volume 2 Exercise 7.1 Samacheer Kalvi
⇒ p2 – 1 = 1
⇒ p2 = 1 + 1 = 2
p = ± \(\sqrt{2}\)
-31 – q3 = -4
-q3 = -4 + 31 = 27
q3 = -27 = (-3)3
⇒ q = -3
r + 1 = \(\frac{3}{2}\)
⇒ r = \(\frac{3}{2}\) – 1 = \(\frac{3-2}{2}\) = \(\frac{1}{2}\)
s – 1 = π
⇒ s = – π + 1 (i.e.,) s = 1 – π
So, p = ± \(\sqrt{2}\), q = -3, r = 1/2 and s = 1 – π

11th Maths 7.1 Exercise Question 3.
Determine the value of x + y if 11th Maths Exercise 7.1 Tamil Medium Samacheer Kalvi
Solution:
\(\left[\begin{array}{cc}{2 x+y} & {4 x} \\ {5 x-7} & {4 x}\end{array}\right]=\left[\begin{array}{cc}{7} & {7 y-13} \\ {y} & {x+6}\end{array}\right]\)
⇒ 2x + y = 7 ………….. (1)
4x = 7y – 13 ………….. (2)
5x – 7 = y …………… (3)
4x = x + 6 ……………. (4)
from (4) 4x – x = 6
3x = 6 ⇒ x = \(\frac{6}{3}\) = 2
Substituting x = 2 in (1), we get
2(2) + y = 7 ⇒ 4 + y = 7 ⇒ y = 7 – 4 = 3
So x = 2 and y = 3
∴ x + y = 2 + 3 = 5

11th Maths Exercise 7.1 Question 4.
Determine the matrices A and B if they satisfy
11th Maths Matrices And Determinants Pdf Samacheer Kalvi
Solution:
Matrices And Determinants Class 11 State Board Solutions Samacheer Kalvi
11th Maths Matrix And Determinants Samacheer Kalvi
11th Maths Volume 2 Exercise 7.1 Answers Samacheer Kalvi

11th Maths Matrix Solutions Question 5.
If A = \(\left[\begin{array}{ll}{\mathbf{1}} & {\boldsymbol{a}} \\ {\mathbf{0}} & {\mathbf{1}}\end{array}\right]\), then compute A4
Solution:
11 Matric Maths Solutions Samacheer Kalvi
Class 11th Maths Exercise 7.1 Samacheer Kalvi

11th Maths Volume 2 Exercise 7.1 Question 6.
Consider the matrix Aα = \(\left[\begin{array}{cc}{\cos \alpha} & {-\sin \alpha} \\ {\sin \alpha} & {\cos \alpha}\end{array}\right]\)
(i) Show that AαAβ = Aα + β.
(ii) Find all possible real values of satisfying the condition Aα + ATα = 1.
Solution:
11th Maths Determinants Solutions Samacheer Kalvi
General solution is α = 2nπ + \(\frac{\pi}{3}\), n ∈ Z

11th Maths Exercise 7.1 Tamil Medium Question 7.
If A = \(\left[\begin{array}{rr}{4} & {2} \\ {-1} & {x}\end{array}\right]\) such that (A – 2I) (A – 3I) = 0, find the value of x.
Solution:
Matrices And Determinants Class 11 Solutions Pdf Samacheer Kalvi

11th Maths Matrices And Determinants Pdf Question 8.
If A = \(\left[\begin{array}{ccc}{\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} \\ {\boldsymbol{a}} & {\boldsymbol{b}} & {-\mathbf{1}}\end{array}\right]\), show that A2 is a unit matrix.
Solution:
Class 11 Matric Maths Solutions Samacheer Kalvi

Matrices And Determinants Class 11 State Board Solutions Question 9.
If A = Class 11 Maths Ex 7.1 Solutions Samacheer Kalvi and A3 – 6A2 + 7A + KI = 0, find the value of k.
Solution:
11th Matrices And Determinants Samacheer Kalvi
Class 11th Maths Chapter 7 Exercise 7.1 Samacheer Kalvi

11th Maths Matrix And Determinants Question 10.
Give your own examples of matrices satisfying the following conditions in each case:
(i) A and B such that AB ≠ BA.
(ii) A and B such that AB = 0 = BA, A ≠ 0 and B ≠ 0.
(iii) A and B such that AB = 0 and BA ≠ 0.
Solution:
11th Maths Book Samacheer Kalvi Chapter 7 Exercise 7.1
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 21
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 22

11th Maths Volume 2 Exercise 7.1 Answers Question 11.
Show that f(x) f(y) = f(x + y), where f(x) = Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 23
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 24
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 25

11 Matric Maths Solutions Question 12.
If A is a square matrix such that A2 = A, find the value of 7A – (I + A)3.
Solution:
Given A2 = A
So 7A – (I + A)3 = 7A – (I + 3A + 3A2 + A3]
= 7A – I – 3A – 3 A2 – A3
Given A2 = A
7A – I – 3A – 3A – A3 = -I + A – A3
= -I + A – (A2 × A)
= -I + A – (A × A) = -I + A – A2
= -I + A – A = -I
So the value of 7A – (I + A)3 = -I.

Class 11th Maths Exercise 7.1 Question 13.
Verify the property A (B + C) = AB + AC, when the matrices A, B and C are given by
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 26
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 27
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 28

11th Maths Determinants Solutions Question 14.
Find the matrix A which satisfies the matrix relation Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 29
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 30
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 31

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1

Matrices And Determinants Class 11 Solutions Pdf Question 15.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 32
(i) (A + B)T = AT + BT = BT + AT
(ii) (A – B)T = AT – BT
(iii) (BT)T = B.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 33
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 34
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 35
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 36

Class 11 Matric Maths Solutions Question 16.
If A is a 3 × 4 matrix and B is a matrix such that both ATB and BAT are defined, what is the order of the matrix B?
Sol.
A is a matrix of order 3 × 4
So AT will be a matrix of order 4 × 3
AT B will be defined when B is a matrix of order 3 × n
BAT will be defirted when B is of order m × 4
from (1) and (2) we see that B should be a matrix of order 3 × 4

Class 11 Maths Ex 7.1 Solutions Question 17.
Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 37
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 38
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 39
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 40

11th Matrices And Determinants Question 18.
Find the matrix A such that Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 41
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 42
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 43

Class 11th Maths Chapter 7 Exercise 7.1 Question 19.
If A = Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 44 is a matrix such that AAT = 9I, find the values of x and y.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 45

11th Maths Book Question 20.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 46
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 47
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 48

Question 21.
Construct the matrix A = [aij]3×3, where aij = i- j. State whether A is symmetric or skew- symmetric.
Solution:
Given A is a matrix of order 3 × 3
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 49
Here AT = -A
⇒ A is skew symmetric

Question 22.
Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.
Solution:
Let A and B be two symmetric matrices
⇒ AT = A and BT = B …………….. (1)
Given that AB = BA (2)
To prove AB is symmetric:
Now (AB)T = BTAT = BA
(from(1)) But (AB)T = AB by ………….. (2)
⇒ AB is symmetric.
Conversely let AB be a symmetric matrix.
⇒ (AB)T = AB
i.e. BTAT = AB
i.e. BA = AB (from (1))
⇒ AB is symmetric

Question 23.
If A and B are symmetric matrices of same order, prove that
(i) AB + BA is a symmetric matrix.
(li) AB – BA is a skew-symmetric matrix.
Solution:
Given A and B are symmetric matrices
⇒ – AT = A and BT = B
(i) To prove AB + BA is a symmetric matrix.
Proof: Now (AB + BA)T = (AB)T + (BA)T = BTAT + ATBT
= BA + AB = AB + BA
i.e. (AB + BA)T = AB + BA
⇒ (AB + BA) is a symmetric matrix.
(ii) To prove AB – BA is a skew symmetric matrix.
Proof: (AB – BA)T = (AB)T – (BA)T = BTAT – ATBT = BA – AB
i.e. (AB – BA)T = – (AB – BA)
⇒ AB – BA is a skew symmetric matrix.

Question 24.
A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins and almonds.
Pack I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds. Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds. Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds. The cost of 50 gm of cashew nuts is ₹ 50, 50 gm of raisins is ₹ 10, and 50 gm of almonds is₹ 60. What is the cost of each gift pack?
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 50
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 51

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 Additional Problems

Question 1.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 52
Prove that (i) AB ≠ BA
(ii) A(BC) = (AB) C
(iii) A(B + C) = AB + AC
(iv) AI = IA = A
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 53
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 54
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 55
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 56

Question 2.
If A = \(\left[\begin{array}{ll}{2} & {3} \\ {4} & {5}\end{array}\right]\) find A2 – 7A – 2I.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 57
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 58

Question 3.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 59
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 60

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 61

Question 4.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 62
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 63
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 64
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 65

Question 5.
If A = \(\left[\begin{array}{rr}{3} & {-5} \\ {-4} & {2}\end{array}\right]\), show that A2 – 5A – 14I = 0 where I is the unit matrix of order 2.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 66

Question 6.
If A = \(\left[\begin{array}{rr}{3} & {-2} \\ {4} & {-2}\end{array}\right]\), find k so that A2 = kA – 2I
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 67
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 68

Question 7.
If A = \(\left[\begin{array}{lll}{1} & {2} & {2} \\ {2} & {1} & {2} \\ {2} & {2} & {1}\end{array}\right]\), show that A2 – 4A – 5I = 0
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 69
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 70

Question 8.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 71
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 72
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 73
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 74
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 75

Question 9.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 76
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 77
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 78

Question 10.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 79
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 80
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.1 81