MATH 240 Homework assignment 2 January 20, 2015

Maker sure you can answer all of the True-False review questions at the end of sections 2.3

through 2.6 of the textbook. Also, make sure you can do the core problems (section 2.3: 7, 15,

17; section 2.4: 6, 15, 21; section 2.5: 3, 10, 17, 22, 24, 35, 49 and section 2.6: 9, 13, 17, 18, 29, 31,

38).

Then write up solutions to the following to hand in on Tuesday January 27:

1. Use row reduction of the appropriate augmented matrix to solve the following systems of equations

or to show that no solution exists:

(a)5x 3y 3z 13x 2y 2z 12x y 2z 8

(b)3x 2y z 6x 2y 3z 6

2. (a) What is the rank of the matrix

A

1 23 4

?

What is kernel of this matrix? For which vectors b

b1b2

is it possible to solve Ax b?

(b)What is the rank of the matrix

A

1 2 34 5 67 8 9

?

What is its kernel? For which vectors b

b1b2b3

is it possible to solve Ax b?

(c) More generally, what is the rank of the matrix

A

1 2 3 nn 1 n 2 n 3 2n

......

.... . .

...pn 1qn 1 pn 1qn 2 pn 1qn 3 n2

?

What is its kernel? For which vectors b

b1b2...bn

is it possible to solve Ax b?

23. Use row reduction to find the inverse of the general 2-by-2 matrix

a bc d

. Dont worry about

whether various quantities might be zero if you have to divide by them (like dividing the first row

by a to get a 1 in the upper left corner). At the end of the whole process, you should find that all

of the entries in A1 have the same denominator. What is it?

4. If A is not a square matrix, say A has m rows and n columns with m n, then we say that the

n-by-m matrix B is a left inverse for A if the product BA is the n-by-n identity matrix. Similarly,

we say that the n-by-m matrix C is a right inverse for A if the product AC is the m-by-m identity

matrix.

Explain how you could use a method similar to the one we used in class for square matrices to

find the left or right inverse of a matrix (if one exists) and use it to find at least two right inverses

of the matrix

1 4 2 32 0 1 03 0 1 2

and at least two left inverses of the matrix

5 2 41 3 00 1 00 2 4

.