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MATH 1502 Study Guide - Quiz Guide: Diagonalizable Matrix, Invertible Matrix

1 pages91 viewsFall 2012

Department
Mathematics
Course Code
MATH 1502
Professor
Geronimo
Study Guide
Quiz

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Math 1502 Practice Test 5 Geronimo
1a. Find the eigenvalues and corresponding eigenvectors to the matrix
A=
1 3
3 1
b. Find A15 (Show all work).
c. Solve ¯xn=A¯xn1,¯x(0) =
1
2
.
d. Suppose Bis a 4 ×4 matrix with three distinct eigenvalues. One eigenvalue has
geometric multiplicity one and one has geometric multiplicity two. Is it possible that
Bis not diagonalizable?
2. Let Lbe the line in R3given by
L={¯x=t
1
2
1
,−∞ < t < ∞}
and ¯y= [2,1,1]T
a. Find the orthogonal projection of ¯yonto L.
b. Find the matrix representing the orthogonal projection onto L.
c. Find the matrix that represents the reflection of a vector about L.
3a. Two matrices Aand Bare similar if B=P1AP where Pis an invertible matrix.
Show that Aand Bhave the same determinant.
3b. Show that Aand Bhave the same characteristic polynomial.
3c. Find the characteristic polynomial of the matrix
A=
102
112
032
.
4(a) Let Abe an m×nmatrix. Show NullACol(AT)
(b) Let,
¯v1= [1,0,1,1]T,¯v2= [1,1,1,0]T,¯v3= [1,2,1,1]T.
From the above vectors construct an orthonormal set of vectors.
(c) Let S be the space spanned by ¯v1,¯v2and ¯v3. If ¯x= [1,1,1,1]Tfind the vector
in S closest to ¯x.
(d) Let Abe the matrix whose columns are ¯v1,¯v2,¯v3and Qbe the matrix whose
columns are the orthonormal basis just constructed. Find Rso that A=QR.
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