Department

Mathematics

Course Code

MATH 1502

Professor

Geronimo

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¯xn−1,¯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=P−1AP 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 NullA⊥Col(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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