MATH 225 - Bilkent University



MATH 225

Exercise Sheet

1)

Consider the following matrices:

A = [pic], B = [pic], C = [pic], D = [pic],

E = [pic], and F = [pic].

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 = [pic]

and B = [pic].

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 = [pic] 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) = [pic].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)[pic]0.

7) Compute the trace of each of the following matrices.

a) A = [pic], b) B = [pic], c) C = [pic]

9) Let A = [pic].

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 = [pic].

12) Determine all 2x2 matrices A such that AB = BA for any 2x2 matrix B.

13) Find a 2x2 matrix B[pic]02 and B[pic]I2 such that AB = BA, where A = [pic].

How many such matrices B are there?

14) A square matrix A is called symmetric if A[pic]=A. It is called skew-symmetric if

A[pic]= -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 [pic]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 = [pic] . 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

[pic], where x ≤ 0, with the usual operations in R[pic].

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[pic]?

a)W1 = {(x,y,x+2y)| x,y[pic]R}

b)W2 = { ( x,y,0)|x,y[pic]R}

c)W3 = {(x,y,z)|x,y,z[pic] R and x+ 2y-z = 0 }

25) Which of the given subsets are sub-spaces of M23?

a) W = { [pic]| b = a+c }

b) W = { [pic]| c> 0}

c) W = { [pic]| 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[pic] is a sub-space of R[pic] ?

b. Show that a line l in R[pic] not passing through the origin is not a sub-space of R[pic].

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[pic].

29) Which of the following sets of vectors span R[pic] ?

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 =[pic]

31) Does the set { [pic]} span M22 ? Are these vectors

linearly independent over M22 ?

32) Are the following vectors linearly independent in P2 ?

a) x[pic] + 1, x - 2, x + 3

b) 2x[pic] + x + 1, 3x[pic] + x - 5, x+13

33) Let [pic] and [pic] be in R[pic]

a) Show that [pic] and [pic] are subspaces of R[pic].

b) Find [pic][pic][pic] and describe the geometric configuration which corresponds to [pic][pic][pic].

c) Show that [pic][pic][pic] is a subspace of R[pic].

34) Let [pic] and [pic] be two subspaces of the vector space V.

a) Show that [pic] and [pic] are subspaces of V.

b) Is W[pic][pic] a subspace of V? Explain your answer.

35) Let [pic], [pic], [pic] and [pic].

a) Find the span { [pic]}.

b) Is w in span { [pic]}?

36) Let [pic], [pic].Determine if [pic]is a basis for R[pic]?

37) Let [pic][pic][pic][pic].Find a basis for the subspace

W spanned by [pic].

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