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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

State the conditions for a system of linear equations to have (i) a unique solution (ii) infinitely many solutions (iii) no solution. Illustrate each with an example using the rank method.

linear-systemsrank
2long10 marks

Define a symmetric and a skew-symmetric matrix. Show that every square matrix can be expressed uniquely as the sum of a symmetric and a skew-symmetric matrix.

matrixsymmetric
3long10 marks

Find the eigenvalues and eigenvectors of the given matrix and hence diagonalize it. Use the diagonal form to compute (A^4).

eigenvaluesdiagonalization
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Define a skew-symmetric matrix with an example.

matrix
5short5 marks

Find the determinant of [[2,1,0],[1,3,1],[0,1,2]].

determinant
6short5 marks

Define the image and kernel of a linear map.

linear-transformation
7short5 marks

What is the algebraic multiplicity of an eigenvalue?

eigenvalues
8short5 marks

Define an inner product on (R^n).

inner-product
9short5 marks

State whether the vectors (1,2), (2,4) are linearly independent.

linear-independence
10short5 marks

Define a finite-dimensional vector space.

dimension
11short5 marks

What is the canonical form of a quadratic form?

quadratic-form
12short5 marks

Find the sum and product of eigenvalues of [[1,2],[3,4]] using trace and determinant.

eigenvalues