CSIT 2nd Semester
Mathematics II Board Question Paper 2082 Old Course

When the system of equations is consistent and inconsistent? Give an example of each. Determine \( x, y, z \) if the system
\[ 2x - 3y + 5z = 5 \] \[ -x + 4y - 2z = 1 \] \[ 3x + 2y - 7z = -15 \] is consistent.

Define inverse of a matrix. Find the inverse of a matrix \( A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix} \), if it exists.

Find the least square solutions of \( Ax = b \) with \( A = \begin{bmatrix} 1 & -2 \\ -1 & 2 \\ 0 & 3 \\ 2 & 5 \end{bmatrix}, b = \begin{bmatrix} 3 \\ 1 \\ -4 \\ 2 \end{bmatrix}. \)

What do you mean by LU factorization? Find LU factorization of
\[\begin{bmatrix} 1 & 2 & 5 \\ 6 & 3 & 8 \\ -5 & 2 & 3 \end{bmatrix}.\]

Let a linear transformation \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be defined by \( T(x) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \). Find the images under \( T \) of \( u = \begin{bmatrix} 2 \\ -3 \end{bmatrix}, v = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \) and \( T(u + v) \).

When two column vectors in \(\mathbb{R}^2\) are equal? Give an example. Compute \(3u - v\) and
\[v - 2u, \text{ where } u = \begin{bmatrix} 2 \\ -3 \\ 1 \end{bmatrix}, v = \begin{bmatrix} 3 \\ -1 \\ 1 \end{bmatrix}. \]

Find the eigenvalues and eigenvectors for the matrix
\[\begin{bmatrix} 1 & 3 \\ 1 & -1 \end{bmatrix}. \tag{2+3}\]

Let \(u = (2, 3, -2, 0)\). Find a unit vector in the same direction as \(u\).

$$ A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 4 & 7 \\ 0 & 0 & 5 \end{bmatrix} $$

Let \(A = \begin{bmatrix} 2 & 5 \\ -3 & 1 \end{bmatrix}\) and \(B = \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix}\). What value(s) of \(k\), if any will make \(AB = BA\)?

Show that the vectors \(u = \begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}, v = \begin{bmatrix} 2 \\ 5 \\ 9 \end{bmatrix}, w = \begin{bmatrix} -3 \\ 9 \\ 3 \end{bmatrix}\) are linearly dependent.