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If the eigen values of a 4 x 4 matrix A are 2, 4, 7, -3 then the rank of a matrix A is
- a)4
- b)0
- c)3
- d)2
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If the eigen values of a 4 x 4 matrix A are 2, 4, 7, -3 then the rank...
Given that the eigen values of A are 2, 4, 7, -3
⇒ all the eigen values of A are non-zero number but we know that
IAI = product of eigen values of A
⇒ IAI = (2)(4)(7)(-3) ≠ 0
⇒ IAI ≠ 0
∴ The rank of a matrix A4x4 is 4, ( since if IAnxnI ≠ 0 then ρ(Anxn) = n)
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If the eigen values of a 4 x 4 matrix A are 2, 4, 7, -3 then the rank...
Rank of a matrix:
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by its rows or columns.
Given information:
The eigenvalues of matrix A are 2, 4, 7, and -3.
Eigenvalues and rank:
The rank of a matrix is equal to the number of non-zero eigenvalues of the matrix. In this case, since all the eigenvalues are non-zero, the rank of matrix A will be equal to the number of eigenvalues.
Calculation:
The given matrix A is a 4x4 matrix and has 4 eigenvalues. Therefore, the rank of matrix A will be 4.
Conclusion:
The correct answer is option 'A' - 4.
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by its rows or columns.
Given information:
The eigenvalues of matrix A are 2, 4, 7, and -3.
Eigenvalues and rank:
The rank of a matrix is equal to the number of non-zero eigenvalues of the matrix. In this case, since all the eigenvalues are non-zero, the rank of matrix A will be equal to the number of eigenvalues.
Calculation:
The given matrix A is a 4x4 matrix and has 4 eigenvalues. Therefore, the rank of matrix A will be 4.
Conclusion:
The correct answer is option 'A' - 4.
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If the eigen values of a 4 x 4 matrix A are 2, 4, 7, -3 then the rank of a matrix A isa)4b)0c)3d)2Correct answer is option 'A'. Can you explain this answer?
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