Algebra Ex 3.18-10th Std Maths-Book Back Question And Answer
Question 1.
Find the order of the product matrix AB if
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.
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
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:
Order of matrix AB = 3 × 3
Order of matrix AB = 4 × 2
Order of matrix AB = 4 × 2
Order of matrix AB = 4 × 1
Order of matrix AB = 1 × 3
Question 4.
find AB, BA and check if AB = BA?
Answer:
Question 5.
Given that
verify that A(B + C) = AB + AC
Answer:
From (1) and (2) we get
A (B + C) = AB + AC
Question 6.
Show that the matrices
satisfy commutative property AB = BA
Answer:
From (1) and (2) we get
AB = BA. It satisfy the commutative property.
Question 7.
Show that (i) A(BC) = (AB)C
(ii) (A-B)C = AC – BC
(iii) (A-B)T = AT – BT
Answer:
From (1) and (2) we get
A(BC) = (AB)C
From (1) and (2) we get
(A – B) C = AC – BC
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From (1) and (2) we get
(A-B)T = AT – BT
Question 8.
then snow that A2 + B2 = I.
Answer:
Question 9.
prove that AAT = I.
Answer:
AAT = I
∴ L.H.S. = R.H.S.
Question 10.
Verify that A2 = I when
Answer:
∴ L.H.S. = R.H.S.
Question 11.
show that A2 – (a + d)A = (bc – ad)I2.
Answer:
L.H.S. = R.H.S.
A2 – (a + d) A = (bc – ad)I2
Question 12.
verify that (AB)T = BT AT
Answer:
From (1) and (2) we get, (AB)T = BT AT
Question 13.
show that A2 – 5A + 7I2 = 0
Answer:
L.H.S. = R.H.S.
∴ A2 – 5A + 7I2 = 0
