Group  ^{List No.}^{Last Name}^{Name}  ^{Student No.}^{Department}  ^{Section}^{Signature} ^{:}^{:}  ^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{Code}^{Acad. Year}^{Semester}^{Instructor}^{Date}^{Time}^{Duration}  M E T U Department of Mathematics^{Basic Linear Algebra}^{Midterm I}^{Math 260}^{1999-2000}^{Fall}^{M.T,S.K,S.F,S..}^{13.11.1999}^{10:00}^{100 minutes}  ^{4 QUESTIONS ON 4 PAGES}^{TOTAL 100 POINTS}^{1}  ^{2}  ^{3}  ^{4}

\underline{Q1.}

(10 points) a) Find k and s such that the following two matrices A and B are \underline{row-equivalent}

A = [ \begin{matrix}1 & k & 0 & 0 & −1 \\ −1 & −k & 1 & 3 & 3 \\ 2 & 2k & 0 & 0 & −2 \end{matrix} ] , B = [ \begin{matrix}0 & 0 & s & 3 & 2 \\ 1 & 2 & s & 3 & 1 \\ 1 & 2 & 2s & 6 & 3 \end{matrix} ]

(15 points) b) Find an invertible matrix P such that P A = B.

\underline{Q2.}

(15 points) a) Find the solution of the following system by using \underline{reduction procedure}

u + u + 2w = 2
2u + 3u − w = 5
3u + 4u + 3w = 7

(10 points) b) Write down a 2 × 2 \underline{non-homogeneous} system with infinitely many solutions.

\underline{Q3.}

(10 points) a) Compute the following determinants ( a ≠ 0, b ≠ 0, a ≠ −b)

i)  | \begin{matrix}a & 3 + a & a \\ a & a & a \\ b + a & b + a & b + a \end{matrix} |  ii)  | \begin{matrix}a & 3 + a & a \\ a & a & a \\ b + a & 3 + b + a & a \end{matrix} |

(15 points) b) Let A be a 4×4 real matrix with det(A) = 5 and B be a 4×4 matrix such that AB = det(A).I_{4}. Compute the followings

i) det(A^{T}) ii) det(B) iii) det(3A) iv) det(AB)

v) det(2A + A).

\underline{Q4.} Let A be a skew-symmetric matrix (A = −A^{T})

A = [ \begin{matrix}0 & 1 & 2 & 1 \\ −1 & 0 & 0 & 0 \\ −2 & 0 & 0 & 1 \\ −1 & 0 & −1 & 0 \end{matrix} ]

(10 points) a) Find the determinant of A. Is A invertible? (15 points) b) Show that every real 3 × 3 skew-symmetric matrix is \underline{not invertible}.