Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18

Students can download Maths Chapter 3 Algebra Ex 3.18 Questions and Answers, Notes, Samacheer Kalvi 10th Maths Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus, helps students complete homework assignments and to score high marks in board exams.

Tamilnadu Samacheer Kalvi 10th Maths Solutions Chapter 3 Algebra Ex 3.18

Question 1.
Find the order of the product matrix AB if
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 1
Answer:
Given A = [aij]p×q and B = [aij]q×r
Order of product of AB = p × r
Order of product of BA is not defined. Number columns in r is not equal to the number of rows in P.
∴ Product BA is not defined.

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18

Question 2.
A has ‘a’ rows and ‘a + 3 ’ columns. B has ‘6’ rows and ‘17 – b’ columns, and if both products AB and BA exist, find a, b?
Solution:
A has a rows, a + 3 columns.
B has b rows, 17 – b columns
If AB exists a × a + 3
b × 17 – b
a + 3 = 6 ⇒ a – 6 = -3 ………… (1)
If BA exists 6 × 17-6
a × a + 3
17 – 6 = a ⇒ a + 6 = 17 …………. (2)
(1) + (2) ⇒ 2a = 14 ⇒ a = 7
Substitute a = 7 in (1) ⇒ 7 – b = -3 ⇒ b = 10
a = 7, b = 10

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18

Question 3.
A has ‘a’ rows and ‘a + 3 ’ columns. B has rows and ‘b’ columns, and if both products AB and BA exist, find a,b?
Answer:

  1. Order of matrix AB = 3 × 3
  2. Order of matrix AB = 4 × 2
  3. Order of matrix AB = 4 × 2
  4. Order of matrix AB = 4 × 1
  5. Order of matrix AB = 1 × 3

Question 4.
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 2
find AB, BA and check if AB = BA?
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 3

Question 5.
Given that
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 4
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 5
verify that A(B + C) = AB + AC
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 6

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 7
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 8
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 9
From (1) and (2) we get
A (B + C) = AB + AC

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18

Question 6.
Show that the matrices
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 99Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 10
satisfy commutative property AB = BA
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 11
From (1) and (2) we get
AB = BA. It satisfy the commutative property.

Question 7.
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 12
Show that (i) A(BC) = (AB)C
(ii) (A-B)C = AC – BC
(iii) (A-B)T = AT – BT
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 13
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 14
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 15
From (1) and (2) we get
A(BC) = (AB)C

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 16

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 17

From (1) and (2) we get
(A – B) C = AC – BC
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 18
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 19
From (1) and (2) we get
(A-B)T = AT – BT

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18

Question 8.
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 20
then snow that A2 + B2 = I.
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 21

Question 9.
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 22
prove that AAT = I.
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 23
AAT = I
∴ L.H.S. = R.H.S.

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18

Question 10.
Verify that A2 = I when
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 24
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 25
∴ L.H.S. = R.H.S.

Question 11.
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 26
show that A2 – (a + d)A = (bc – ad)I2.
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 27
L.H.S. = R.H.S.
A2 – (a + d) A = (bc – ad)I2

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18

Question 12.
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verify that (AB)T = BT AT
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 29
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 30
From (1) and (2) we get, (AB)T = BT AT

Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18

Question 13.
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show that A2 – 5A + 7I2 = 0
Answer:
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 32
Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.18 33
L.H.S. = R.H.S.
∴ A2 – 5A + 7I2 = 0

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