CSIT 2nd Semester
Mathematics II Board Question Paper 2082

Define homogeneous linear system of equations with an example. Which type of homogeneous equation has a nontrivial solution? Determine the value of \( x, y \), and \( z \) if the system of equation \( x - 2y + 3z = 0 \), \( -x + 2y - 4z = 0 \), \( 2x - 4y + 9z = 0 \) has a nontrivial solution.

Let \( T \) be defined by \( T(x) = Ax \), where \( A = \begin{bmatrix} 1 & -5 & -7 \\ -3 & 7 & 5 \end{bmatrix} \). Find a vector \( x \) whose image under \( T \) is \( b \), where \( b = \begin{bmatrix} -2 \\ -2 \end{bmatrix} \), and determine whether \( x \) is unique or not.

(a) Find the basis and dimension of the subspace \( H = \left\{ \begin{bmatrix} a - 3b + 6c \\ 5a + 4d \\ b - 2c - d \\ 5d \end{bmatrix} : a, b, c, d \in \mathbb{R} \right\} \).
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(b) What is rank of a matrix? Find the rank of a matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{bmatrix} \).

Let \( A = \begin{bmatrix} 1 & 2 \\ 3 & 0 \end{bmatrix} \). Find \( A^3 \).

Let \( A = \begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix} \). Find \( 5|A| - 2|I| \). Is det(5A) equal to 5 det \( A^2 \) Justify.

Define subspace of a vector space. Prove that \( H = \left\{ \begin{bmatrix} a \\ 0 \\ c \end{bmatrix} : a, c \in \mathbb{R} \right\} \) is a subspace of \(\mathbb{R}^3\).

Let \( x = \begin{bmatrix} 7 \\ 6 \end{bmatrix} \) and \( u = \begin{bmatrix} 4 \\ 2 \end{bmatrix} \). Find the orthogonal projection of \( x \) onto \( u \). Also, write \( x \) as the sum of two orthogonal vectors, one in span \(\{u\}\) and one orthogonal to \( u \).

Find LU factorization of \( A = \begin{bmatrix} 3 & 2 \\ -2 & 3 \end{bmatrix} \).

Use Cramer's rule to solve the equations \( \frac{4}{x} + \frac{6}{y} = 4 \) and \( \frac{3}{x} - \frac{4}{y} = \frac{1}{6} \).

Does \( (\mathbb{Z},.) \) form a group? Justify.

If \( R \) is a ring with additive identity 0, then prove that for any \( a, b \in R \)
(i) \( 0a = a0 = 0 \)
(ii) \( (-a)(-b) = ab \).