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For a skew symmetric odd ordered matrix A of integers, which of the following will hold true:
- a)det(A) = 9
- b)det(A) = 81
- c)det(A) = 0
- d)et(A) = 4
Correct answer is option 'C'. Can you explain this answer?
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For a skew symmetric odd ordered matrix A of integers, which of the fo...
Determinant of a skew symmetric odd ordered matrix A is always 0 .
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For a skew symmetric odd ordered matrix A of integers, which of the fo...
Skew Symmetric Matrix:
A skew symmetric matrix is a square matrix whose transpose is equal to its negative. In other words, for a matrix A to be skew symmetric, it must satisfy the condition A^T = -A.
Odd Ordered Matrix:
An odd-ordered matrix is a matrix whose number of rows and columns is odd.
Determinant of a Matrix:
The determinant of a matrix is a scalar value that can be calculated for square matrices. It represents certain properties of the matrix and is denoted by det(A).
Explanation:
For a skew symmetric matrix A of odd order, we can make some observations about its determinant:
1. Determinant of a skew symmetric matrix of odd order is always 0:
- Since the matrix is skew symmetric, A^T = -A.
- Taking determinant on both sides, we have det(A^T) = det(-A).
- Using the property of determinants, we know that det(A^T) = det(A) and det(-A) = (-1)^n * det(A), where n is the order of the matrix.
- Therefore, det(A) = (-1)^n * det(A).
- Since n is odd, (-1)^n = -1.
- So, det(A) = -det(A).
- The only value that satisfies this equation is det(A) = 0.
2. Hence, the correct answer is option 'C' (det(A) = 0).
Summary:
For a skew symmetric matrix A of odd order, the determinant will always be 0.
A skew symmetric matrix is a square matrix whose transpose is equal to its negative. In other words, for a matrix A to be skew symmetric, it must satisfy the condition A^T = -A.
Odd Ordered Matrix:
An odd-ordered matrix is a matrix whose number of rows and columns is odd.
Determinant of a Matrix:
The determinant of a matrix is a scalar value that can be calculated for square matrices. It represents certain properties of the matrix and is denoted by det(A).
Explanation:
For a skew symmetric matrix A of odd order, we can make some observations about its determinant:
1. Determinant of a skew symmetric matrix of odd order is always 0:
- Since the matrix is skew symmetric, A^T = -A.
- Taking determinant on both sides, we have det(A^T) = det(-A).
- Using the property of determinants, we know that det(A^T) = det(A) and det(-A) = (-1)^n * det(A), where n is the order of the matrix.
- Therefore, det(A) = (-1)^n * det(A).
- Since n is odd, (-1)^n = -1.
- So, det(A) = -det(A).
- The only value that satisfies this equation is det(A) = 0.
2. Hence, the correct answer is option 'C' (det(A) = 0).
Summary:
For a skew symmetric matrix A of odd order, the determinant will always be 0.
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