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Chapter 3: Problem 70

If you think of numbers as \(1 \times 1\) matrices, which numbers are invertible \(1 \times 1\) matrices?

Short Answer

Expert verified
In conclusion, every non-zero number can be considered as an invertible \(1 \times 1\) matrix, while zero is not invertible because it does not have a multiplicative inverse.

Step by step solution

01

Understand invertible 1x1 matrices

A 1x1 matrix is simply a scalar value, and we need to determine which scalar values are invertible. In order for a 1x1 matrix to be invertible, it must have a multiplicative inverse, which means when it is multiplied by its inverse, the result should be the identity matrix. In this case, our "identity matrix" is simply the number 1, since we are dealing with scalar values.
02

Find the multiplicative inverse for a given number

The multiplicative inverse of a number 'a' is a number 'b' such that their multiplication results in 1 (i.e., a 脳 b = 1). To determine if a given number has a multiplicative inverse, we can check if its inverse (1/a) multiplied by the original number gives us 1.
03

Find invertible numbers

Zero does not have a multiplicative inverse because dividing by zero is undefined. Non-zero numbers do have a multiplicative inverse (1/a) because when multiplied together (a 脳 (1/a)), they result in 1. So, all non-zero numbers can be considered as invertible 1x1 matrices.
04

Conclusion

In conclusion, every non-zero number can be considered as an invertible 1x1 matrix, while zero is not invertible because it does not have a multiplicative inverse.

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Most popular questions from this chapter

Campaign Strategies \(^{28}\) Florida and Ohio are "swing states" that have a large bounty of electoral votes and are therefore highly valued by presidential campaign strategists. Suppose it is now the weekend before Election Day 2008 , and each candidate (McCain and Obama) can visit only one more state. Further, to win the election, McCain needs to win both of these states. Currently McCain has a \(40 \%\) chance of winning Ohio and a \(60 \%\) chance of winning Florida. Therefore, he has a \(0.40 \times 0.60=0.24\), or \(24 \%\) chance of winning the election. Assume that each candidate can increase his probability of winning a state by \(10 \%\) if he, and not his opponent, visits that state. If both candidates visit the same state, there is no effect. a. Set up a payoff matrix with McCain as the row player and Obama as the column player, where the payoff for a specific set of circumstances is the probability (expressed as a percentage) that McCain will win both states. b. Where should each candidate visit under the circumstances?

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Translate the given matrix equations into svstems of linear equations. $$ \left[\begin{array}{lrll} 0 & 1 & 6 & 1 \\ 1 & -5 & 0 & 0 \end{array}\right]\left[\begin{array}{r} x \\ y \\ z \\ w \end{array}\right]=\left[\begin{array}{r} -2 \\ 9 \end{array}\right] $$

A diagonal matrix \(D\) has the following form. $$ D=\left[\begin{array}{ccccc} d_{1} & 0 & 0 & \ldots & 0 \\ 0 & d_{2} & 0 & \ldots & 0 \\ 0 & 0 & d_{3} & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & d_{n} \end{array}\right] $$ When is \(D\) singular? Why?
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