Mathematics II TU Board 2076 question paper

When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x – 2y = 5, -x + y + 5z = 2, y + z = 0

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.

Find the least-square solution of Ax=b for $A=\left[\begin{array}{cc}1& -6\\ 1& -2\\ 1& 1\\ 1& 7\end{array}\right]$ and $b=\left[\begin{array}{c}-1\\ 2\\ 1\\ 6\end{array}\right]$

Let T is a linear transformation. Find the standard matrix of T such that

1. T:R^2 → R^4 by T(e1) = (3, 1, 3, 1) and T(e_2) = (-5, 2, 0, 0) where e_1 = (1, 0) and e_2 = (0, 1);
2. T:R^2 → R^4 rotates point as the origin through $\frac{3\pi }{4}$ radians counter clockwise.
3. T:R^2 → R^4 Is a vertical shear transformation that maps e_1 into e_1-2e_2 but leaves vector e_2 unchanged.

For what value of h will y be in span {v_1 , v_2, v_3} if $v_1=\left[\begin{array}{c}1\\ -1\\ -2\end{array}\right]$, $v_2=\left[\begin{array}{c}5\\ -4\\ -7\end{array}\right]$, $v_3=\left[\begin{array}{c}-3\\ 1\\ 0\end{array}\right]$ and $y=\left[\begin{array}{c}-4\\ 3\\ h\end{array}\right]$

Let us define a linear transformation T:R^2 → R^2by T(x) = $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$ = $\left[\begin{array}{c}-x_2\\ x_1\end{array}\right]$. Find the image under T of $u=\left[\begin{array}{c}4\\ 1\end{array}\right]$, $v=\left[\begin{array}{c}2\\ 3\end{array}\right]$ and u + v = $\left[\begin{array}{c}6\\ 4\end{array}\right]$

Let $A=\left[\begin{array}{cc}2& 5\\ -3& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}4& -5\\ 3& k\end{array}\right]$. Determine the value (s) of k if any will make AB = BA.

Define determinant. Compute the determinant without expanding $\left[\begin{array}{ccc}-2& 8& -9\\ -1& 7& 0\\ 1& -4& 2\end{array}\right]$

Define null space . Find the basis for the null space of the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\end{array}\right]$

Let B = {b_1, b_2} and C = (c_1, c_2) be bases for a vector V, and suppose b_1 = -c_1 + 4c_2 and b_2 = 5c_1 – 3c_2. Find the change of coordinate matrix for a vector space and find [x]_c for x = 5b_1 + 3b_2.

Find the eigen values of the matrix $\left[\begin{array}{cc}6& 5\\ -8& -6\end{array}\right]$

Find the QR factorization of the matrix $\left[\begin{array}{cc}2& 1\\ 3& -1\end{array}\right]$

Define binary operation. Determine whether the binary operation * is associative or commutative or both where * is defined on Q by letting $x\ast y=\frac{x+y}{3}$

Show that the ring (Z_4, +_4, ._4) is an integral domain.

Find the vector x determined by the coordinate vector $[x{]}_{\beta }=\left[\begin{array}{c}-4\\ 8\\ 7\end{array}\right]$ where $\beta =\left\{\left[\begin{array}{c}-1\\ 2\\ 0\end{array}\right],\left[\begin{array}{c}3\\ -5\\ 2\end{array}\right],\left[\begin{array}{c}4\\ -7\\ 3\end{array}\right]\right\}$