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Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?
- a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.
- b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.
- c)If P and Q are invertible, then [QP]-1 = P-1Q-1.
- d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.
Correct answer is option 'A,C'. Can you explain this answer?
Most Upvoted Answer
Pand Q are two square matrices of the same order. Which of the followi...
Answer:
Statement a: If P and Q are invertible, then [PQ]-1 = Q-1P-1.
Statement c: If P and Q are invertible, then [QP]-1 = P-1Q-1.
To determine the correctness of these statements, let's consider the properties of invertible matrices and matrix multiplication.
Invertible matrices:
- An invertible matrix, also known as a non-singular matrix, is a square matrix that has an inverse.
- If a matrix A is invertible, then A-1 is the inverse of A, such that A * A-1 = A-1 * A = I, where I is the identity matrix.
- Invertible matrices have the property that their determinant is non-zero.
Matrix multiplication:
- Matrix multiplication is not commutative, i.e., AB ≠ BA in general.
- However, for invertible matrices A and B, AB and BA are both invertible, and their inverses are related as (AB)-1 = B-1A-1 and (BA)-1 = A-1B-1.
Now, let's analyze each statement:
Statement a: If P and Q are invertible, then [PQ]-1 = Q-1P-1.
This statement is correct. According to the properties of matrix multiplication and invertible matrices, we have:
[PQ]-1 = Q-1P-1 (Using the property (AB)-1 = B-1A-1)
Statement c: If P and Q are invertible, then [QP]-1 = P-1Q-1.
This statement is also correct. By applying the same property of matrix multiplication and invertible matrices, we have:
[QP]-1 = P-1Q-1
Therefore, both statements a and c are correct.
Conclusion:
- If P and Q are invertible matrices, then both [PQ]-1 = Q-1P-1 and [QP]-1 = P-1Q-1 hold true.
- In general, the order of multiplication matters in matrix multiplication, and the inverses of matrices should be multiplied in the reverse order to obtain the inverse of the product matrix.
Statement a: If P and Q are invertible, then [PQ]-1 = Q-1P-1.
Statement c: If P and Q are invertible, then [QP]-1 = P-1Q-1.
To determine the correctness of these statements, let's consider the properties of invertible matrices and matrix multiplication.
Invertible matrices:
- An invertible matrix, also known as a non-singular matrix, is a square matrix that has an inverse.
- If a matrix A is invertible, then A-1 is the inverse of A, such that A * A-1 = A-1 * A = I, where I is the identity matrix.
- Invertible matrices have the property that their determinant is non-zero.
Matrix multiplication:
- Matrix multiplication is not commutative, i.e., AB ≠ BA in general.
- However, for invertible matrices A and B, AB and BA are both invertible, and their inverses are related as (AB)-1 = B-1A-1 and (BA)-1 = A-1B-1.
Now, let's analyze each statement:
Statement a: If P and Q are invertible, then [PQ]-1 = Q-1P-1.
This statement is correct. According to the properties of matrix multiplication and invertible matrices, we have:
[PQ]-1 = Q-1P-1 (Using the property (AB)-1 = B-1A-1)
Statement c: If P and Q are invertible, then [QP]-1 = P-1Q-1.
This statement is also correct. By applying the same property of matrix multiplication and invertible matrices, we have:
[QP]-1 = P-1Q-1
Therefore, both statements a and c are correct.
Conclusion:
- If P and Q are invertible matrices, then both [PQ]-1 = Q-1P-1 and [QP]-1 = P-1Q-1 hold true.
- In general, the order of multiplication matters in matrix multiplication, and the inverses of matrices should be multiplied in the reverse order to obtain the inverse of the product matrix.
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Community Answer
Pand Q are two square matrices of the same order. Which of the followi...
∵ P and Q are not invertible, so
[PQ]-1 ≠ Q-1 P-1
If P and Q are invertible then,
[OP]-1 = P-1 Q-1
If P and Q are invertible, then
[PQ]-1 = Q-1 P-1.
[PQ]-1 ≠ Q-1 P-1
If P and Q are invertible then,
[OP]-1 = P-1 Q-1
If P and Q are invertible, then
[PQ]-1 = Q-1 P-1.
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Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer?
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Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer?.
Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer?.
Solutions for Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Civil Engineering (CE).
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Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer?, a detailed solution for Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer? has been provided alongside types of Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Pand Q are two square matrices of the same order. Which of the following statement(s) is/are correct?a)If P a n d Q are invertible, then [PQ]-1 = Q-1P-1.b)If P and Q are invertible, then [PQ]-1 = P-1Q-1.c)If P and Q are invertible, then [QP]-1 = P-1Q-1.d)If P and Q are not invertible, then [PQ]-1 = Q-1P-1.Correct answer is option 'A,C'. Can you explain this answer? tests, examples and also practice Civil Engineering (CE) tests.
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