Define determinant. Evaluate without expanding: [5]

Find the eigenvalues of the matrix [5]

Define group. Show that the set of all integers, Z forms group under addition operation. [5]

When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve:
[10]

What is the condition of a matrix to have an inverse? Find the inverse of the matrix [10]

Change into reduce echelon form of the matrix [5]

Let A and B be matrices. What value(s) of k if any will make AB = BA? [5]

Find LU factorization of the matrix [5]

Find the least-square solution of Ax = b for [10]

State and prove the Pythagorean theorem of two vectors and verify this for u = (1, -1) and v = (1, 1). [5]

Define ring with an example. Compute the product in the given ring (-3,5), (2,-4) in $Z_4$ x $Z_{11}$. [5]

Define linear transformation with an example. Is a transformation T(x, y) = (3x + y, 5x + 7y, x+3y) linear? Justify. [5]

Find the dimension of the null space and column space of $$A = \begin{bmatrix} -3 & 6 & -1 & 1 & -7 \\ 1 & -2 & 2 & 3 & -1 \\ 2 & -4 & 5 & 8 & -4 \end{bmatrix}$$ [5]

Define linearly independent set of vectors with an example. Show that the vectors (1,4,3), (0,3,1) and (3,-5,4) are linearly independent. Do they form a basis? Justify. [10]

Define subspace of a vector space. Let $ H = {\begin{bmatrix} s \\ t \\ O \end{bmatrix} : s, t \in \mathbb{R}}$, show that H is a subspace of $\mathbb{R}^3$. [5]