Mathematics II TU Board 2075 question paper

When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve: x + y + z = 4, x + 2y + 2z = 2, 2x + 2y + z = 5.

What is the condition of a matrix to have an inverse? Find the inverse of the matrix $A=\left[\begin{array}{ccc}5& 1& 2\\ 1& 0& 3\\ 4& -3& 8\end{array}\right]$ If it exists.

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.

Find the least-square solution of Ax = b for A = $\left[\begin{array}{ccc}1& 3& 5\\ 1& 1& 0\\ 1& 1& 2\\ 1& 3& 3\end{array}\right]$ and b= $\left(\begin{array}{c}3\\ 5\\ 7\\ 3\end{array}\right)$

Change into reduce echelon form of the matrix $\left(\begin{array}{ccc}0& 3& -6\\ 3& -7& 8\\ 3& -9& 12\end{array}\right)$.

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

Let $A=\left(\begin{array}{cc}-1& -2\\ 5& 9\end{array}\right)$ and $b=\left(\begin{array}{cc}9& 2\\ k& -1\end{array}\right)$. What value (s) of k if any will make AB = BA?

Define determinant. Evaluate without expanding $\left|\begin{array}{ccc}1& 5& -6\\ -1& -4& 4\\ -2& -7& 9\end{array}\right|$

Define subspace of a vector space. Let $H=\left\{\left(\begin{array}{c}s\\ t\\ 0\end{array}\right):s,t\in R\right\}$. Show that H is a subspace of:

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

Find the eigenvalues of the matrix $\left(\begin{array}{ccc}6& 3& -8\\ 0& -2& 0\\ 1& 0& -3\end{array}\right)$

Find LU factorization of the matrix \begin{pmatrix}2 & 5\ 6 & -7\end{pmatrix}

Define group. Show that the set of all integersZ forms group under addition operation.

Define ring with an example. Compute the product in the given ring (-3, 5) (2, -4) in Z_4 x Z_11.

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