# Infinitely Many Solutions? A Homework Problem to Investigate

• blinder
In summary, the conversation discusses a homework problem involving a true or false question about a singular n by n matrix and its solutions. It is noted that the question is unclear and may have multiple answers depending on the value of b.
blinder
I had a homework problem and i was wondering if anyone could help me out with it. It is a true or fasle question that requires explanation whether true or false. It goes like this...If A is a singular n by n matrix, then A*x=b has infinitely many solutions.(True or False)

Is the matrix $N = (0)$ singular? Does $Nx = b$ have infinitely many solutions if b is nonzero?

There are several case :

b=0, then Ax=0 has infinitely many solution if det(A)=0

b<>0 then it can have zero or an infinity of solution, depending on the determinants : $$det(b|_nA)$$

where $$b|_nA$$ means : the nth column of A is replace by b.

So the answer is "true and false" because you don't specify enough the question.

or you might say the question is meaningless since you do not quantify the letter "b".

## 1. What is meant by "infinitely many solutions" in this homework problem?

In this context, "infinitely many solutions" refers to a mathematical equation or problem that has an infinite number of possible solutions, rather than a finite or limited number.

## 2. How do you approach solving a problem with infinitely many solutions?

The first step is to identify the problem and its given parameters, such as equations or constraints. Then, you can use mathematical techniques such as substitution or graphing to find solutions. However, since there are infinitely many solutions, it may be more useful to find a general solution or pattern rather than a specific one.

## 3. Can you give an example of a problem with infinitely many solutions?

One example is the equation x + y = 5, where there are infinitely many possible combinations of x and y that would satisfy the equation, such as (2,3), (1,4), or (0,5).

## 4. Are there any real-life applications of problems with infinitely many solutions?

Yes, many real-life problems can have infinitely many solutions. For example, in economics, the demand and supply functions for a product can have infinitely many equilibrium points where the quantity demanded and supplied are equal. In physics, certain problems involving infinite series or continuous functions may have infinitely many solutions.

## 5. How can understanding problems with infinitely many solutions be useful in scientific research?

Problems with infinitely many solutions can provide valuable insights into the nature of mathematical equations and systems. By studying these problems, scientists can develop new techniques and methods for solving them, which can then be applied to other areas of research. Additionally, understanding infinitely many solutions can help scientists better understand and model complex systems in nature that may have a large number of variables and possible outcomes.

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