VIPSolutionsA is a square matrix, and B is an m × n matrix. Check the true statements below: A. The determinant of A is the product of the diagonal entries in A. B. det(-A) = - det (A). C. If two row interchanges are made in succession, then the determinant of the new matrix is equal to the determinant of the original matrix. D. det(BBT) = det(B) det(BT) E. If A is a 5 × 5 skew symmetric matrix, then the system AX = 0 has nontrivial solutions. F. det(A + I) = det(A) + 1. G. If det(A) = 0, then two rows or two columns are the same, or a row or a column is zero.
Question:
A is a square matrix, and B is an m × n matrix. Check the true statements below: A. The determinant of A is the product of the diagonal entries in A. B. det(-A) = - det (A). C. If two row interchanges are made in succession, then the determinant of the new matrix is equal to the determinant of the original matrix. D. det(BBT) = det(B) det(BT) E. If A is a 5 × 5 skew symmetric matrix, then the system AX = 0 has nontrivial solutions. F. det(A + I) = det(A) + 1. G. If det(A) = 0, then two rows or two columns are the same, or a row or a column is zero.
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Asked by: MD ISRARUL HAQUE
Created at: 2025-07-27 09:00:35
Asked by: MD ISRARUL HAQUE
Created at: 2025-07-27 09:00:35
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Let's analyze each statement one by one:
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**A. The determinant of A is the product of the diagonal entries in A.**
- **This is true only if A is a triangular matrix (upper or lower triangular), not for all square matrices.**
- **General Case:** False.
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**B. det(-A) = - det(A).**
- For an \( n \times n \) matrix, \( \det(-A) = (-1)^n \det(A) \).
- **Only true if n is odd. Not true in general.**
- **General Case:** False.
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**C. If two row interchanges are made in succession, then the determinant of the new matrix is equal to the determinant of the original matrix.**
- Each row interchange multiplies the determinant by -1. Two successive interchanges multiply by \((-1)^2 = 1\).
- So the determinant is unchanged.
- **True.**
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**D. det(B B^T) = det(B) det(B^T)**
- Only defined if \( B \) is square (i.e., \( m = n \)). For a general \( m \times n \) matrix, \( B B^T \) is \( m \times m \), \( B \) is \( m \times n \), \( B^T \) is \( n \times m \), and determinants are not defined unless \( m = n \).
- **In general, False.**
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**E. If A is a 5 × 5 skew symmetric matrix, then the system AX = 0 has nontrivial solutions.**
- For any odd \( n \), the determinant of a skew-symmetric \( n \times n \) matrix is zero, so the system has nontrivial solutions.
- **True.**
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**F. det(A + I) = det(A) + 1.**
- This is **false** in general. There is no such property.
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**G. If det(A) = 0, then two rows or two columns are the same, or a row or a column is zero.**
- **False.** The determinant can be zero for many reasons, not just these (e.g., rows are linearly dependent without being equal or zero).
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## **Summary Table**
| Statement | True or False | Reason |
|:---|:---|:---|
| A | False | Only true if triangular. |
| B | False | Only true if n is odd. |
| C | **True** | Two swaps multiply by 1. |
| D | False | Only if B is square. |
| E | **True** | Odd-dim skew-symmetric has det 0. |
| F | False | Not a property of determinants. |
| G | False | Zero det means linear dependence, not necessarily identical/zero rows. |
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## **Final Answer:**
**C and E are true.**
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