###### Elementary Linear Algebra

238 SOLVED PROBLEMS

Question: Appendix.2

## Using Mathematical Induction Use mathematical induction to prove the following formula. Sn = 1 + 3 + 5 + 7 + ··· + (2n – 1) = n² ...

Mathematical induction consists of two distinct pa...
Question: Appendix.4

## Using Mathematical Induction in Linear Algebra If A1, A2, · · ·, An are invertible matrices, then prove the generalization of Theorem 2.9. (A1 A2 A3 · · · An)^-1 = An^-1 · · · A3^-1A2^-1A1^-1  ...

1.  The formula is valid trivially when n = 1 beca...
Question: Appendix.3

## Using Mathematical Induction Use mathematical induction to prove the formula for the sum of the first n squares. Sn = 1² + 2² + 3² + 4² + · · · + n² = n(n + 1)(2n + 1) / 6 ...

1.  When n = 1, the formula is valid, because [lat...
Question: Appendix.5

## Using Proof by Contradiction Prove that √2 is an irrational number. ...

Begin by assuming that $\sqrt{2}$  is...
Question: Appendix.1

## Sum of Odd Integers Use the pattern to propose a formula for the sum of the first n odd integers. 1 = 1  1 + 3 = 4  1 + 3 + 5 = 9 1 + 3+ 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25  ...

Notice that the sums on the right are equal to the...
Question: Appendix.6

## Using Proof by Contradiction A positive integer greater than 1 is a prime when its only positive factors are 1 and itself. Prove that there are infinitely many prime numbers. ...

Assume there are only finitely many prime numbers,...
Question: Appendix.7

## Using Proof by Contradiction in Linear Algebra Let A and B be n × n matrices such that AB is singular. Prove that either A or B is singular. ...

Assume that neither A or B is singular. Because yo...
Question: Appendix.8

## Using a Counterexample Use a counterexample to show that the statement is false. Every odd number is prime. ...

Certainly, you can list many odd numbers that are ...
Question: Appendix.9

## Using a Counterexample in Linear Algebra Use a counterexample to show that the statement is false. If A and B are square singular matrices of order n, then A + B is a singular matrix of order n. ...

Let A = \begin{bmatrix} 1 & 0 \\ 0 &...