Math 260 Basic Linear Algebra Midterm Exam I (Spring 1999-2000)

1 / (25)  2 / (25)  3 / (30)  4 / (20)  TOTAL

4 questions on 4 pages Total 100 points

Name and Surname: Student No: Section: Signature: Duration: 110 minutes

1. Let A = ( \begin{matrix}1 & 2 & 4 & 1 \\ 1 & 2 & 8 & 1 \\ 2 & 2 & 4 & 2 \\ 2 & 2 & 8 & 3 \end{matrix} ) ∈ ℝ^{4×4}. Find the inverse of A by using elementary

row operations. (Write all the elementary row operations used).

2. Given the homogeneous system

x_{2} + x_{3} + 2x_{5} − 3x_{6} = 0
2x_{1} + 4x_{3} + 6x_{5} + 4x_{6} = 0
−2x_{1} − 4x_{3} + x_{4} − 5x_{5} = 0
2x_{2} + 2x_{3} + 4x_{5} − 6x_{6} = 0

(i) find the row-reduced echelon form of the coefficient matrix of this system. (Write all the elementary row operations used) (ii) find the general solution of this system. (iii) find the fundamental solution(s) of this system.

3. Let A = ( \begin{matrix}0 & 1 & 0 & 2 \\ 1 & 2 & 3 & 4 \\ 2 & 0 & 1 & 0 \\ 1 & 2 & 3 & 0 \end{matrix} ) ∈ ℝ^{⋭×⋭} and let B be the matrix obtained by

applying the following elementary row operations to A in the given order: 2R_{1}, 2R_{4}, 3R_{2}, \frac{1}{2}R_{1} + R_{3}, R_{2} ↔ R_{3}.

a) Compute the following determinants:

(i) | A | (vi) | A adj (A) | (ii) | A^{−2} | (vii) | B | (iii) | −A | (viii) | AB | (iv) | 2A + A | (ix) | R | where R is the row-reduced

echelon matrix obtained from A.

(v) | adj (cA) |, (c ∈ ℝ)

(b) Compute [adj (A)]_{32} = The (3, 2)-entry of adj (A)

4. (i) Let A, B, P, Q be four matrices satisfying P A − QB = 0. If P and Q are invertible matrices show that A and B are row-equivalent.

(ii) Let A be a 27 × 27 real matrix. If A is a skew-symmetric matrix then show that det(A) = 0.