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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Define rank of a matrix. Find the rank of the matrix by reducing it to echelon form, and solve a system of linear equations using the Gauss elimination method.

matrixranklinear-systems
2long10 marks

State and prove the Cayley-Hamilton theorem. Use it to find the inverse of a given 3x3 matrix.

matrixcayley-hamilton
3long10 marks

Define eigenvalues and eigenvectors. Find the eigenvalues and corresponding eigenvectors of a given 3x3 matrix.

eigenvalueseigenvectors
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Define a symmetric matrix and an orthogonal matrix with examples.

matrix
5short5 marks

Find the rank of the matrix [[1,2,3],[2,4,6],[1,1,1]].

rank
6short5 marks

State the properties of determinants.

determinant
7short5 marks

What is a null space of a matrix?

vector-space
8short5 marks

Define a basis of a vector space.

basis
9short5 marks

Find the eigenvalues of the matrix [[2,0],[0,3]].

eigenvalues
10short5 marks

Show that the vectors (1,0) and (0,1) are linearly independent.

linear-independence
11short5 marks

Define adjoint of a matrix.

matrix
12short5 marks

What is a singular matrix?

matrix