CSIT 2nd Semester
Mathematics II Board Question Paper 2081

Define augmented matrix with an example. Find the general solution of the linear system whose augmented matrix is \[ \begin{bmatrix} 1 & 6 & 2 & -5 \\ -1 & 0 & 3 & 1 \\ 0 & -1 & -2 & 3 \end{bmatrix}. \]

(a) Let \( A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \), and define \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) by \( T(x) = Ax \). Find the image under \( T \) of \( u = \begin{bmatrix} 1 \\ -3 \end{bmatrix}, v = \begin{bmatrix} a \\ b \end{bmatrix} \).
(b) Prove that a map \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) defined by \( T(x, y) = (y, x) \) is linear.

Define inverse of a matrix. Find the inverse of a matrix \( A = \begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & 2 \\ -3 & 4 & 8 \end{bmatrix} \), if it exists.

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

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

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

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 \( x = -3b_1 + 2b_2 \).

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

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

Prove that the binary operation * defined on \( Z \) by letting \( m * n = m + n + 1 \) is
commutative and associative.

Show that \((Q, +, .)\) forms a ring.