Math 310: Hour Exam 1 (Solutions) Prof. S. Smith: Mon 28 Sept 1998

Problem 1: (a) Using Gauss-Jordan, find the row-reduced echelon form of the following augmented matrix. (INDICATE the row operations you use).

(A|b) = ( \begin{matrix}1 & 2 & 3 & 6 \\ 0 & 1 & 2 & 3 \\ 2 & 3 & 4 & 9 \end{matrix} ) .

→^{A_{3}^{−2×1}} ( \begin{matrix}1 & 2 & 3 & 6 \\ 0 & 1 & 2 & 3 \\ 0 & −1 & −2 & −3 \end{matrix} ) →^{A_{1}^{−2×2},A_{3}^{1×2}} ( \begin{matrix}1 & 0 & −1 & 0 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \end{matrix} )

(b) Then give the solutions of the corresponding linear system Ax = b.

So solutions are (r, 3 − 2r, r) for free variable x_{3} = r.

Problem 2: (a) What elementary row OPERATION will change

A = ( \begin{matrix}1 & 2 \\ 2 & 1 \end{matrix} ) to B = ( \begin{matrix}1 & 2 \\ 0 & −3 \end{matrix} ) ?

A_{2}^{−2×1} : add −2 times 1st row to 2nd.

(b) What elementary row MATRIX E will, by left muliplication, perform the same operation? (that is, EA = B)

E = ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} ) : (obtained by doing that operation to the identity matrix).

Problem 3: (a) Find the inverse (any method) of:

A = ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{matrix} ) .

Using (A|I) method: use A_{2}^{−1×1}, A_{3}^{−1×1} to clear first column; then A_{3}^{−1×2} to clear second column.

Get inverse: ( \begin{matrix}1 & 0 & 0 \\ −1 & 1 & 0 \\ 0 & −1 & 0 \end{matrix} )

(b) Give the LU-decomposition of

A = ( \begin{matrix}1 & 2 \\ 1 & 3 \end{matrix} );

that is, find lower-triangular L and upper-triangular U, so that A = LU.

Apply A_{2}^{−1×1} to get U = ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ). So L from inverse operation A_{2}^{+1×1} is ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )

Problem 4: (a) Find the determinant of

A = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{matrix} ) .

Top row: 1(1.1 − 0.1) − 1(0.1 − 1.1) + 0 = (1) − (−1) = 2.

(b) Use Cramer’s rule to solve

( \begin{matrix}1 & 2 & 4 \\ 3 & 4 & 10 \end{matrix} )

det(A) = 1.4 − 3.2 = −2, so x_{1} = −\frac{1}{2}(4.4 − 10.2) = 2 and x_{2} = −\frac{1}{2}(1.10 − 3.4) = 1.

Problem 5: (a) Is (1, 0, 1) in the span of (1, 1, 1) and (1, 2, 1)? Either give coefficients in a linear combination, or explain why it is not possible.

Yes: Set up augmented matrix (A|b) = ( \begin{matrix}1 & 1 & 1 \\ 1 & 2 & 0 \\ 1 & 1 & 1 \end{matrix} ) .

Row-reduction quickly produces coefficients 2 and −1.

(b) Let V be the space of 2 × 2 matrices, and W the subSET of diagonal matrices. Show that W is a subSPACE of V.

A ∈ W says diagonal, meaning 0 off the diagonal, so A has form ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ).

Similarly if B ∈ W it has form ( \begin{matrix}c & 0 \\ 0 & d \end{matrix} )

Is A + B ∈ W? It is ( \begin{matrix}a + c & 0 \\ 0 & b + d \end{matrix} ), also diagonal, so yes.

For scalar r, is rA ∈ W? It is ( \begin{matrix}ra & 0 \\ 0 & rb \end{matrix} ), also diagonal, so yes.

We see “yes”, W is a subspace of V.