Class 12 Exam > Class 12 Questions > Which of the following is not a property of i...
Start Learning for Free
Which of the following is not a property of invertible matrices if A and B are matrices of the same order?
- a)(AB)-1 = A-1 B-1
- b)(AA-1) = (A-1 A) = I
- c)(AB)-1 = B-1 A-1
- d)AB = BA = I
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Which of the following is not a property of invertible matrices if A a...
Properties of Invertible Matrices:
1. (AA-1) = (A-1 A) = I
2. AB = BA = I
3. (A-1)-1 = A
4. (kA)-1 = (1/k)A-1, where k is a non-zero scalar
5. (AB)-1 = B-1 A-1
Explanation:
Option A states that (AB)-1 = A-1 B-1, which is not a property of invertible matrices. This statement is false because in general, (AB)-1 ≠ A-1 B-1.
To see why this is true, consider the case where A and B are both 2x2 matrices:
A = [a b]
[c d]
B = [e f]
[g h]
Then AB is given by:
AB = [ae+bg af+bh]
[ce+dg cf+dh]
The inverse of AB, assuming it exists, is given by:
(AB)-1 = 1/det(AB) [dh -bh -cf af]
[-dg ae ce -af]
where det(AB) = (ae+bg)(cf+dh)-(af+bh)(ce+dg)
On the other hand, the inverse of A and B are given respectively by:
A-1 = 1/det(A) [d -b]
[-c a]
B-1 = 1/det(B) [h -f]
[-g e]
where det(A) = ad-bc and det(B) = eh-fg
Now, if we compute A-1 B-1, we get:
A-1 B-1 = 1/det(A)det(B) [(dh-bgf-eh+fg) -(bh-ae-hf+cf)]
[(-dg+ce+bg-af) (ag-ce-df+ae)]
Notice that A-1 B-1 is not equal to (AB)-1 in general, unless A and B commute (i.e. AB = BA).
Therefore, option A is not a property of invertible matrices.
1. (AA-1) = (A-1 A) = I
2. AB = BA = I
3. (A-1)-1 = A
4. (kA)-1 = (1/k)A-1, where k is a non-zero scalar
5. (AB)-1 = B-1 A-1
Explanation:
Option A states that (AB)-1 = A-1 B-1, which is not a property of invertible matrices. This statement is false because in general, (AB)-1 ≠ A-1 B-1.
To see why this is true, consider the case where A and B are both 2x2 matrices:
A = [a b]
[c d]
B = [e f]
[g h]
Then AB is given by:
AB = [ae+bg af+bh]
[ce+dg cf+dh]
The inverse of AB, assuming it exists, is given by:
(AB)-1 = 1/det(AB) [dh -bh -cf af]
[-dg ae ce -af]
where det(AB) = (ae+bg)(cf+dh)-(af+bh)(ce+dg)
On the other hand, the inverse of A and B are given respectively by:
A-1 = 1/det(A) [d -b]
[-c a]
B-1 = 1/det(B) [h -f]
[-g e]
where det(A) = ad-bc and det(B) = eh-fg
Now, if we compute A-1 B-1, we get:
A-1 B-1 = 1/det(A)det(B) [(dh-bgf-eh+fg) -(bh-ae-hf+cf)]
[(-dg+ce+bg-af) (ag-ce-df+ae)]
Notice that A-1 B-1 is not equal to (AB)-1 in general, unless A and B commute (i.e. AB = BA).
Therefore, option A is not a property of invertible matrices.
Free Test
FREE
| Start Free Test |
Community Answer
Which of the following is not a property of invertible matrices if A a...
(AB)-1 = A-1 B-1 is incorrect. The correct formula is (AB)-1 = B-1 A-1. B-1 A-1 ≠ A-1 B-1 as matrix multiplication is not commutative.
|
Explore Courses for Class 12 exam
|
|
Similar Class 12 Doubts
Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer?
Question Description
Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer?.
Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer?.
Solutions for Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 12.
Download more important topics, notes, lectures and mock test series for Class 12 Exam by signing up for free.
Here you can find the meaning of Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer?, a detailed solution for Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Which of the following is not a property of invertible matrices if A and B are matrices of the same order?a)(AB)-1 = A-1 B-1b)(AA-1) = (A-1 A) = Ic)(AB)-1 = B-1 A-1d)AB = BA = ICorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice Class 12 tests.
|
|
Explore Courses for Class 12 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup