Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Let X be a square matrix. Consider the follow...
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Let X be a square matrix. Consider the following two statements on X.
I. X is invertible
II. Determine of X is non-zero.
Which one of the following is TRUE?
I. X is invertible
II. Determine of X is non-zero.
Which one of the following is TRUE?
- a)I implies II; II does not imply I
- b)I does not imply II; II does not imply I
- c)I and II are equivalent statements
- d)II implies I; I does not imply II
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let X be a square matrix. Consider the following two statements on X.I...
If | A | ≠ 0 then A is invertible matrix
If A -1 exists then | A | ≠ 0
∴ I and II are equivalent statements.
If A -1 exists then | A | ≠ 0
∴ I and II are equivalent statements.
Most Upvoted Answer
Let X be a square matrix. Consider the following two statements on X.I...
Understanding the Statements
In the context of square matrices, we analyze the two statements regarding matrix \( X \):
- **Statement I**: \( X \) is invertible.
- **Statement II**: The determinant of \( X \) is non-zero.
Relationship Between Invertibility and Determinant
- **Invertibility**: A square matrix \( X \) is invertible if there exists a matrix \( Y \) such that \( XY = I \), where \( I \) is the identity matrix.
- **Determinant**: The determinant of a matrix provides a scalar value that indicates whether the matrix is invertible. Specifically:
- If \( \text{det}(X) \neq 0 \), then \( X \) is invertible.
- If \( \text{det}(X) = 0 \), then \( X \) is not invertible.
Proof of Equivalence
- **I implies II**: If \( X \) is invertible, then its determinant must be non-zero. Hence, Statement I guarantees Statement II.
- **II implies I**: Conversely, if the determinant of \( X \) is non-zero, then \( X \) must be invertible. Therefore, Statement II guarantees Statement I.
Conclusion
Since both statements imply each other, they are equivalent. This means:
- **Correct Answer**: Option **C**: I and II are equivalent statements.
This equivalence is foundational in linear algebra, illustrating the strong relationship between a matrix's invertibility and its determinant.
In the context of square matrices, we analyze the two statements regarding matrix \( X \):
- **Statement I**: \( X \) is invertible.
- **Statement II**: The determinant of \( X \) is non-zero.
Relationship Between Invertibility and Determinant
- **Invertibility**: A square matrix \( X \) is invertible if there exists a matrix \( Y \) such that \( XY = I \), where \( I \) is the identity matrix.
- **Determinant**: The determinant of a matrix provides a scalar value that indicates whether the matrix is invertible. Specifically:
- If \( \text{det}(X) \neq 0 \), then \( X \) is invertible.
- If \( \text{det}(X) = 0 \), then \( X \) is not invertible.
Proof of Equivalence
- **I implies II**: If \( X \) is invertible, then its determinant must be non-zero. Hence, Statement I guarantees Statement II.
- **II implies I**: Conversely, if the determinant of \( X \) is non-zero, then \( X \) must be invertible. Therefore, Statement II guarantees Statement I.
Conclusion
Since both statements imply each other, they are equivalent. This means:
- **Correct Answer**: Option **C**: I and II are equivalent statements.
This equivalence is foundational in linear algebra, illustrating the strong relationship between a matrix's invertibility and its determinant.
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