Mathematics II (MTH168) 2081 BS Past Questions & Solutions

Attempt any Eight questions.

1. 4

   Verify that $(AB)^{-1} = B^{-1}A^{-1}$ if $A = \begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$.
2. 5

   Find LU factorization of $\begin{bmatrix} 2 & 5 \\ 6 & -7 \end{bmatrix}$
3. 6

   Compute the determinants by cofactor expansions $\begin{vmatrix} 6 & 0 & 0 & 5 \\ 1 & 7 & 2 & -5 \\ 2 & 0 & 0 & 0 \\ 8 & 3 & 1 & 8 \end{vmatrix}$
4. 7

   Show that the column vectors $u = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$, $v = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}$, $w = \begin{bmatrix} 4 \\ 2 \\ 6 \end{bmatrix}$ and $x = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}$ is in span $\{u,v,w\}$.
5. 8

   Let $B = \{b_{1}, b_{2}\}$ and $C = \{c_{1}, c_{2}\}$ be bases for a vector space V, and suppose $b_{1} = 6c_{1} - 2c_{2}$ and $b_{2} = 9c_{1} - 4c_{2}$ then (a) find the change of coordinates matrix from B to C. (b) find $[x]_{C}$ for $x = -3b_{1} + 2b_{2}$
6. 9

   Let $A = \begin{bmatrix} 1 & 6 \\ 5 & 2 \end{bmatrix}$. Find the eigenvalues and eigenvectors of A.
7. 10

   Determine the least squares error in the least-square solution of $Ax = b$, where $A = \begin{bmatrix} 4 & 0 \\ 0 & 2 \\ 1 & 1 \end{bmatrix}$, $\hat{x} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $b = \begin{bmatrix} 2 \\ 0 \\ 11 \end{bmatrix}$
8. 11

   Prove that the binary operation defined on $\mathbb{Z}$ by letting $m * n = m + n + 1$ is commutative and associative.
9. 12

   Show that $(\mathbb{Q}, +, .)$ forms a ring.