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The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct ?
- a)PQ = I but QP ≠ I
- b)QP = I but PQ ≠ I
- c)PQ = I and QP = I
- d)PQ − QP = I
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The matrix P is the inverse of a matrix Q. If I denotes the identity m...
Given P is inverse of Q
⇒ PQ = QP = I
⇒ PQ = QP = I
Most Upvoted Answer
The matrix P is the inverse of a matrix Q. If I denotes the identity m...
Explanation:
To understand why option 'C' is the correct answer, let's first review the properties of matrix inverses.
Matrix Inverse:
The inverse of a square matrix Q, denoted as Q^(-1), is a matrix such that when it is multiplied by the original matrix Q, the result is the identity matrix I. In other words, Q^(-1) * Q = I.
Now, let's consider the given information that matrix P is the inverse of matrix Q.
Option Analysis:
a) PQ = I but QP ≠ I
This option suggests that PQ equals the identity matrix I, but QP is not equal to the identity matrix. However, this contradicts the definition of matrix inverses, as Q^(-1) * Q should equal I. Therefore, this option is incorrect.
b) QP = I but PQ ≠ I
This option suggests that QP equals the identity matrix I, but PQ is not equal to the identity matrix. Again, this contradicts the definition of matrix inverses, as Q^(-1) * Q should equal I. Therefore, this option is also incorrect.
c) PQ = I and QP = I
This option suggests that both PQ and QP are equal to the identity matrix I. This aligns with the definition of matrix inverses, as Q^(-1) * Q equals I. Therefore, this option is correct.
d) PQ * QP = I
This option suggests that the product of PQ and QP is equal to the identity matrix I. While matrix multiplication is associative, meaning (AB)C = A(BC), it is not necessarily commutative, meaning AB ≠ BA in general. Therefore, this option is incorrect.
Conclusion:
Based on the properties of matrix inverses, the correct answer is option 'c) PQ = I and QP = I'. This aligns with the definition of matrix inverses and ensures that both PQ and QP result in the identity matrix I.
To understand why option 'C' is the correct answer, let's first review the properties of matrix inverses.
Matrix Inverse:
The inverse of a square matrix Q, denoted as Q^(-1), is a matrix such that when it is multiplied by the original matrix Q, the result is the identity matrix I. In other words, Q^(-1) * Q = I.
Now, let's consider the given information that matrix P is the inverse of matrix Q.
Option Analysis:
a) PQ = I but QP ≠ I
This option suggests that PQ equals the identity matrix I, but QP is not equal to the identity matrix. However, this contradicts the definition of matrix inverses, as Q^(-1) * Q should equal I. Therefore, this option is incorrect.
b) QP = I but PQ ≠ I
This option suggests that QP equals the identity matrix I, but PQ is not equal to the identity matrix. Again, this contradicts the definition of matrix inverses, as Q^(-1) * Q should equal I. Therefore, this option is also incorrect.
c) PQ = I and QP = I
This option suggests that both PQ and QP are equal to the identity matrix I. This aligns with the definition of matrix inverses, as Q^(-1) * Q equals I. Therefore, this option is correct.
d) PQ * QP = I
This option suggests that the product of PQ and QP is equal to the identity matrix I. While matrix multiplication is associative, meaning (AB)C = A(BC), it is not necessarily commutative, meaning AB ≠ BA in general. Therefore, this option is incorrect.
Conclusion:
Based on the properties of matrix inverses, the correct answer is option 'c) PQ = I and QP = I'. This aligns with the definition of matrix inverses and ensures that both PQ and QP result in the identity matrix I.
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The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct ?a)PQ = I but QP ≠ Ib)QP = I but PQ ≠ Ic)PQ = I and QP = Id)PQ − QP = ICorrect answer is option 'C'. Can you explain this answer?
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The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct ?a)PQ = I but QP ≠ Ib)QP = I but PQ ≠ Ic)PQ = I and QP = Id)PQ − QP = ICorrect answer is option '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 The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct ?a)PQ = I but QP ≠ Ib)QP = I but PQ ≠ Ic)PQ = I and QP = Id)PQ − QP = ICorrect answer is option '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 The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct ?a)PQ = I but QP ≠ Ib)QP = I but PQ ≠ Ic)PQ = I and QP = Id)PQ − QP = ICorrect answer is option 'C'. Can you explain this answer?.
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