MATH 225
Exercise Sheet
1)
Consider the following matrices:
A = , B = , C = , D = ,
E = , and F = .
1) If possible, compute:
a) DA+B b) AB and BA c) 2C-3E d) CB+D e) AB+D2 .
2) If possible, compute:
a) DA+B b) EC c) CE d) EB+F e) FC+D
3) If possible, compute:
a) AT b) (AB) T c) BTAT d) ( C+E)T e) A(2B) and 2(AB)
4) Find a value of r and a value of s so that ABT = 0 , where A =
and B = .
5) Let A be an mxn matrix and B an nxp matrix. What if anything can you say about the matrix product AB when:
a) A has a column consisting entirely of zeros?
b) B has a row consisting entirely of zeros.
6) If A = is an nxn matrix, then the trace of A, Tr(A), is defined as the sum of all elements on the main diagonal of A , Tr(A) = .Show that
a) Tr(cA) =c Tr(A) where c is a real number.
b) Tr(A+B) = Tr(A) + Tr(B)
c) Tr(AB) = Tr(BA)
d) Tr(AT) = Tr(A)
e) Tr(ATA)0.
7) Compute the trace of each of the following matrices.
a) A = , b) B = , c) C =
9) Let A = .
a) Determine the simple expression for A2.
b) Determine the simple expression for A3.
c) Conjecture the form of a simple expression for Ak.
10) Find a pair of unequal 2x2 matrices A and B such that AB = O2x2.
11) Find two different 2x2 matrices A such that A2 = .
12) Determine all 2x2 matrices A such that AB = BA for any 2x2 matrix B.
13) Find a 2x2 matrix B02 and BI2 such that AB = BA, where A = .
How many such matrices B are there?
14) A square matrix A is called symmetric if A=A. It is called skew-symmetric if
A= -A.
a)Show that if A is any mxn matrix, then AAT and ATA are symmetric.
b) Show that If A is symmetric and invertible then is symmetric.
15) Show that if A is any nxn matrix then:
a) A+AT is symmetric.
b)A-At is skew symmetric.
16) Let A be a symmetric matrix.
Show that A2 is also symmetric.
Show that 2A2 - 3A + I is symmetric.
17) a)Show that if A is an nxn matrix, then A = S+K , where S is symmetric and K is skew-symmetric.
b) Let A = . Find matrices S and K in part a).
18) Use Cramer's rule to solve y without solving for x, z and w.
4x + y + z + w = 6
3x + 7y -z + w = 1
7x + 3y -5z + 8w = -3
x + y + z +2w = 3
19) If a matrix A is invertible, show that A2 is also invertible.
20) Suppose A and B are matrices with B is invertible. If A = B-1AB,show
that AB = BA .
21) Let A and B be 3x3 matrices such that det A = 10 and det B = 12. Find det (AB) ,
det (A4) , det (2B), det (AB)T , det (A)-1, and det ( A-1B-1AB ), det( A+B).
22) Suppose A is a 4x4 matrix and P is an invertible matrix such that
PAP-1 = diag ( 1,-1,-1,-1).
a)Find detA.
b) Find A2.
c)Find A101.
23) The given set together with the operations is not a vector space.
List the properties of the definition that fail to hold:
a)The set of all ordered pairs of real numbers with operations
( x,y ) +( x',y' ) = ( x+x', y+y' ) and r( x,y ) = ( x, ry ).
b)The set of all 2x1 matrices
, where x ≤ 0, with the usual operations in R.
c)The set of all ordered pairs of real numbers with the operations
( x,y,z) +( x',y',z' ) = ( x+x', y+y', z+z' ) and r( x,y,z) = ( x, 1, z ).
24) Which of the following subsets are sub-spaces of R?
a)W1 = {(x,y,x+2y)| x,yR}
b)W2 = { ( x,y,0)|x,yR}
c)W3 = {(x,y,z)|x,y,z R and x+ 2y-z = 0 }
25) Which of the given subsets are sub-spaces of M23?
a) W = { | b = a+c }
b) W = { | c> 0}
c) W = { | a= -2c and f= 2e+d }
26) Which of the following subsets are sub-spaces of the vector space
F = the vector space of all real-valued continuous functions?
a. All constant functions.
b. All functions f such that f(0) = 5.
c. All differentiable functions.
d. All integrable functions.
27) a. Show that a line l through the origin of R is a sub-space of R ?
b. Show that a line l in R not passing through the origin is not a sub-space of R.
28)Suppose A is an nxn matrix and k is a scalarShow that the set of all vectors x such that
Ax = kx is a subspace of R.
29) Which of the following sets of vectors span R ?
a) {(3,2,1,0),(1,2,11),(0,0,0,1)}
b) {1,1,0,0),(1,2,-1,1),(0,0,1,-1),(2,1,2,-1)}
30) Find a basis for the solution space of the homogenous system Ax = 0 where
A =
31) Does the set { } span M22 ? Are these vectors
linearly independent over M22 ?
32) Are the following vectors linearly independent in P2 ?
a) x + 1, x - 2, x + 3
b) 2x + x + 1, 3x + x - 5, x+13
33) Let and be in R
a) Show that and are subspaces of R.
b) Find and describe the geometric configuration which corresponds to .
c) Show that is a subspace of R.
34) Let and be two subspaces of the vector space V.
a) Show that and are subspaces of V.
b) Is W a subspace of V? Explain your answer.
35) Let , , and .
a) Find the span { }.
b) Is w in span { }?
36) Let , .Determine if is a basis for R?
37) Let .Find a basis for the subspace
W spanned by .