Prof. A. Suciu

MTH 1230 -- Linear Algebra -- Fall 1997

Sample Quiz 2

(a)
Find the LU-decomposition of $A=\bmatrix -5 & 3 & 4 \\ 10 & -8 & -9 \\ 15 & 1 & 2 \endbmatrix.$
(b)
Use the LU-decomposition of A from (a) to solve $A\cdot \bmatrix u\\ v\\ w \endbmatrix = \bmatrix -3\\ 9\\ 3 \endbmatrix.$

Consider the matrix $A=\bmatrix -1&3&2\\ 2&4&1\\ 4&3&a\endbmatrix.$

(a)
For which values of a is A invertible?
(b)
Find the inverse of A when a=2.

Consider the matrix $A=\bmatrix 3&4\\ -1&2\endbmatrix.$

(a)
Find elementary matrices E₁, E₂, E₃, and E₄ such that $E_4\cdot E_3 \cdot E_2 \cdot E_1 \cdot A = I$ .
(b)
Use item:a to express A as a product of elementary matrices.
(c)
Use item:a to express A^-1 as a product of elementary matrices.
(d)
Use item:c to find A^-1.

Consider the matrices:
$\begin{displaymath} A=\bmatrix 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \endbmatrix... ...C=\bmatrix 1 & 2 & 3 \\ 4 & 5 & 6 \\ 9 & 12 & 15 \endbmatrix.\end{displaymath}$

Find elementary matrices E₁, E₂, E₃, and E₄ such that

(a)
$E_1\cdot A=B$ .
(b)
$E_2\cdot B=A$ .
(c)
$E_3\cdot A=C$ .
(d)
$E_4\cdot C=A$ .

Let $\vec a=\bmatrix 2\\ -1\\ 5\endbmatrix , \quad \vec b=\bmatrix 1\\ 1\\ -2\endbmatrix .$

(a)
Find the lengths of $\vec a$ and $\vec b$ , and compute the dot product $\vec a \cdot \vec b$ .
(b)
Find the angle between $\vec a$ and $\vec b$ . Are $\vec a$ and $\vec b$ orthogonal?

(c)
Find the projection of $\vec a$ onto $\vec b$ .

(d)
Find the projection of $\vec b$ onto $\vec a$ .

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Created by Alexandru Suciu, Wed Oct 8, 1997

alexsuciu@neu.edu

http://www.math.neu.edu/~suciu/mth1230/1230f97.sq2/1230f97.sq2.html