Question 1: If is a real matrix with trace and if and are eigenvalues of , each with algebraic multiplicity , then the determinant of is equal to
Answer: Out of five eigenvalues of matrix we know four . Let the last eigenvalue is . Then,
Hence the determinant, will be product of eigenvalues
Hence 4 is correct choice.
Question 2: For a positive integer , let denote the vector space of polynomials in one variable with real coefficients and with degree . Consider the map defined by . Then
- is a linear transformation and dim range .
- is a linear transformation and dim range .
- is a linear transformation and dim range .
- is not a linear transformation.
Answer: This transformation replaces each by . Also,
Hence this is a linear transformation, and
Hence the nullity will be zero. By rank nullity theorem, we get
Hence 2 is correct choice.
Question 3: Let be a real matrix of rank . Then the rank of , where deonotes the transpose , is
- exactly
- exactly
- exactly
- at most but not necessarily
Answer: Given that the rank of is , hence the nullity of is . Using theorem,
Nullspace of and are the same.
We get the nullity of is 2. Since is matrix. Hence rank of will be .
Hence 1 is correct choice.
Question 4: Let denote the set of all the prime numbers with the property that the matrix
has an inverse in the field . Then
- is infinite
Answer: The matrix is singular, if it’s determinant will be zero in that field.
The determinant will be zero in only for . For all other primes it will be non-zero.
Hence 4 is correct choice.
Question 5: Consider the quadratic form , where
Then
- has rank .
- for some invertible real matrix .
- for some invertible real matrix .
- for some invertible real matrix .
Answer: Here,
When is orthogonal, and both represent equivalent quadratic forms. Then the problem is reduced to find equivalent bilinear forms.
Question 6: Let be an matrix such that , where is the identity matrix of order . Which of the following statements is false?
- .
- trace rank .
- rank + rank = n.
- The eigenvalus of are each equal to .
Answer: The matrix satisfies the polynomial . Hence minimal polynomial is a factor of . Since this polynomial has linear factors, hence matrix is diagonalizable with eigenvalues or or both. Since then at least eigenvalues will be .
Hence 4 is correct choice.