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The matrix A is 3 × 3 matrix, if the matrix is a singular matrix then which of the following statement is true;
- a)Product of eigen values = det(A)
- b)Det(A) = 0
- c)One of the eigen value is zero
- d)All of these
Correct answer is option 'D'. Can you explain this answer?
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The matrix A is 3 × 3 matrix, if the matrix is a singular matrix then...
For singular matrix, the determinant of matrix is zero
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Also product of eigen values = det(matrix)
∴ product of eigen values will be zero, if at least one eigen value is zero.
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The matrix A is 3 × 3 matrix, if the matrix is a singular matrix then...
Understanding Singular Matrices
A singular matrix is one that does not have an inverse, which occurs when its determinant is zero. For a 3 × 3 matrix A, this leads to several important implications regarding its eigenvalues and determinant.
Key Properties of a Singular Matrix
- Determinant Equals Zero:
- For any singular matrix A, it holds that det(A) = 0. This is a fundamental property that directly defines singular matrices.
- Product of Eigenvalues:
- The determinant of a matrix is equal to the product of its eigenvalues. Therefore, if det(A) = 0, then the product of the eigenvalues must also be zero.
- Presence of Zero Eigenvalue:
- Since the product of the eigenvalues equals zero, at least one of the eigenvalues must be zero. This directly implies that the matrix has a non-trivial kernel (null space), further confirming its singularity.
Conclusion
Given these properties, we can conclude the following for a singular 3 × 3 matrix A:
- a) Product of eigenvalues = det(A): True, but does not independently define singularity.
- b) Det(A) = 0: True, defining property of singular matrices.
- c) One of the eigenvalues is zero: True, a direct consequence of the determinant being zero.
Thus, the correct answer is option 'D' - All of these statements are true for a singular matrix. Each statement reinforces the condition of singularity, affirming the interconnected nature of determinants and eigenvalues in linear algebra.
A singular matrix is one that does not have an inverse, which occurs when its determinant is zero. For a 3 × 3 matrix A, this leads to several important implications regarding its eigenvalues and determinant.
Key Properties of a Singular Matrix
- Determinant Equals Zero:
- For any singular matrix A, it holds that det(A) = 0. This is a fundamental property that directly defines singular matrices.
- Product of Eigenvalues:
- The determinant of a matrix is equal to the product of its eigenvalues. Therefore, if det(A) = 0, then the product of the eigenvalues must also be zero.
- Presence of Zero Eigenvalue:
- Since the product of the eigenvalues equals zero, at least one of the eigenvalues must be zero. This directly implies that the matrix has a non-trivial kernel (null space), further confirming its singularity.
Conclusion
Given these properties, we can conclude the following for a singular 3 × 3 matrix A:
- a) Product of eigenvalues = det(A): True, but does not independently define singularity.
- b) Det(A) = 0: True, defining property of singular matrices.
- c) One of the eigenvalues is zero: True, a direct consequence of the determinant being zero.
Thus, the correct answer is option 'D' - All of these statements are true for a singular matrix. Each statement reinforces the condition of singularity, affirming the interconnected nature of determinants and eigenvalues in linear algebra.
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The matrix A is 3 × 3 matrix, if the matrix is a singular matrix then which of the following statement is true;a)Product of eigen values = det(A)b)Det(A) = 0c)One of the eigen value is zerod)All of theseCorrect answer is option 'D'. Can you explain this answer?
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