(Solved): 1. (a) Define Hermitian and skew-Hermitian matrices. If A is a nonsingular matrix, then show that A ...

1. (a) Define Hermitian and skew-Hermitian matrices. If $A$ is a nonsingular matrix, then show that $A^T$ is nonsingular and ${\left(A^{?1}\right)}^T={\left(A^T\right)}^{?1}$. (b) Use Gauss-Jordan method to find the inverse of the matrix $A$, where $A=?\begin{array}{lll}1& 2& 3\\ 2& 3& 1\\ 4& 3& 3\end{array}?$ 5. (a) If a line makes angles $?,?,?,?$ with the four diagonals of a cube, then find the value of ${\cos }^2?+{\cos }^2?+{\cos }^2?+{\cos }^2?$. (b) Find conditions upon which the two lines whose direction cosines are connected by the relations $al+bm+cn=0$ and $u{l}^2+v{m}^2+w{n}^2=0$ are perpendicular and parallel. 4. (a) Given $A=?\begin{array}{ccccr}1& ?2& 0& 0& 3\\ 2& ?5& ?3& ?2& 6\\ 0& 5& 15& 10& 0\\ 2& 6& 18& 8& 6\end{array}?$. Find bases for the row space and null space of $A$. Also, verify the dimension theorem for $A$. (b) Find an $LU$-decomposition of $A=?\begin{array}{ccc}6& ?2& 0\\ 9& ?1& 1\\ 3& 7& 5\end{array}?$.