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Which of the following statement is false?
- a)Every skew symmetric matrix of odd order is non-singular
- b)If the determinant of a square matrix is non zero, then matrix is non-singular
- c)The adjoint symmetric matrix of symmetric matrix is symmetric
- d)The adjoint matrix of a diagonal matrix is diagonal
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Which of the following statement is false?a)Every skew symmetric matri...
The determinant of skew-symmetric matrix of odd order is always zero
Therefore the inverse of skew-symmetric matrix does not exist
Therefore the inverse of skew-symmetric matrix does not exist
This question is part of UPSC exam. View all JEE courses
Most Upvoted Answer
Which of the following statement is false?a)Every skew symmetric matri...
False statement: Every skew symmetric matrix of odd order is non-singular.
Explanation:
To understand why this statement is false, let's first define what a skew symmetric matrix is and what it means for a matrix to be non-singular.
A skew symmetric matrix is a square matrix where the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal. In other words, for any skew symmetric matrix A, A[i][j] = -A[j][i] for all i and j.
A matrix is said to be non-singular (or invertible) if its determinant is non-zero. The determinant of a matrix is a scalar value that can be calculated using various methods, such as cofactor expansion or row reduction.
Counterexample:
Let's consider a counterexample to prove that the statement is false. Suppose we have a 3x3 skew symmetric matrix A:
A = [0 a b]
[-a 0 c]
[-b -c 0]
where a, b, and c are non-zero real numbers.
Now, let's calculate the determinant of matrix A:
det(A) = 0(a(0) - c(-b)) - a(-a(0) - c(-b)) + b(-a(-b) - c(0))
= 0 + a^2 + b^2
Since a, b, and c are non-zero, the determinant of matrix A is non-zero. Therefore, matrix A is non-singular.
From this counterexample, we can see that there exist skew symmetric matrices of odd order that are non-singular, which contradicts the given statement. Hence, the statement is false.
Conclusion:
The false statement is: Every skew symmetric matrix of odd order is non-singular.
Explanation:
To understand why this statement is false, let's first define what a skew symmetric matrix is and what it means for a matrix to be non-singular.
A skew symmetric matrix is a square matrix where the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal. In other words, for any skew symmetric matrix A, A[i][j] = -A[j][i] for all i and j.
A matrix is said to be non-singular (or invertible) if its determinant is non-zero. The determinant of a matrix is a scalar value that can be calculated using various methods, such as cofactor expansion or row reduction.
Counterexample:
Let's consider a counterexample to prove that the statement is false. Suppose we have a 3x3 skew symmetric matrix A:
A = [0 a b]
[-a 0 c]
[-b -c 0]
where a, b, and c are non-zero real numbers.
Now, let's calculate the determinant of matrix A:
det(A) = 0(a(0) - c(-b)) - a(-a(0) - c(-b)) + b(-a(-b) - c(0))
= 0 + a^2 + b^2
Since a, b, and c are non-zero, the determinant of matrix A is non-zero. Therefore, matrix A is non-singular.
From this counterexample, we can see that there exist skew symmetric matrices of odd order that are non-singular, which contradicts the given statement. Hence, the statement is false.
Conclusion:
The false statement is: Every skew symmetric matrix of odd order is non-singular.
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Which of the following statement is false?a)Every skew symmetric matrix of odd order is non-singularb)If the determinant of a square matrix is non zero, then matrix is non-singularc)The adjoint symmetric matrix of symmetric matrix is symmetricd)The adjoint matrix of a diagonal matrix is diagonalCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Which of the following statement is false?a)Every skew symmetric matrix of odd order is non-singularb)If the determinant of a square matrix is non zero, then matrix is non-singularc)The adjoint symmetric matrix of symmetric matrix is symmetricd)The adjoint matrix of a diagonal matrix is diagonalCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statement is false?a)Every skew symmetric matrix of odd order is non-singularb)If the determinant of a square matrix is non zero, then matrix is non-singularc)The adjoint symmetric matrix of symmetric matrix is symmetricd)The adjoint matrix of a diagonal matrix is diagonalCorrect answer is option 'A'. Can you explain this answer?.
Which of the following statement is false?a)Every skew symmetric matrix of odd order is non-singularb)If the determinant of a square matrix is non zero, then matrix is non-singularc)The adjoint symmetric matrix of symmetric matrix is symmetricd)The adjoint matrix of a diagonal matrix is diagonalCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Which of the following statement is false?a)Every skew symmetric matrix of odd order is non-singularb)If the determinant of a square matrix is non zero, then matrix is non-singularc)The adjoint symmetric matrix of symmetric matrix is symmetricd)The adjoint matrix of a diagonal matrix is diagonalCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statement is false?a)Every skew symmetric matrix of odd order is non-singularb)If the determinant of a square matrix is non zero, then matrix is non-singularc)The adjoint symmetric matrix of symmetric matrix is symmetricd)The adjoint matrix of a diagonal matrix is diagonalCorrect answer is option 'A'. Can you explain this answer?.
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Which of the following statement is false?a)Every skew symmetric matrix of odd order is non-singularb)If the determinant of a square matrix is non zero, then matrix is non-singularc)The adjoint symmetric matrix of symmetric matrix is symmetricd)The adjoint matrix of a diagonal matrix is diagonalCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for Which of the following statement is false?a)Every skew symmetric matrix of odd order is non-singularb)If the determinant of a square matrix is non zero, then matrix is non-singularc)The adjoint symmetric matrix of symmetric matrix is symmetricd)The adjoint matrix of a diagonal matrix is diagonalCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of Which of the following statement is false?a)Every skew symmetric matrix of odd order is non-singularb)If the determinant of a square matrix is non zero, then matrix is non-singularc)The adjoint symmetric matrix of symmetric matrix is symmetricd)The adjoint matrix of a diagonal matrix is diagonalCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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