Linear Algebra problem (linear equations)

In summary, the conditions for no solution in the given augmented matrix are a = 0, 2 or a = 2, b != -1. The conditions for a unique solution are a = 2, b = -1. And if both the numerator and denominator of the solution for x1 and x2 have the form 0/0, there are infinitely many solutions.
  • #1
kirab
27
0

Homework Statement



Given augmented matrix

[tex]\left(\begin{array}{ccc}a&1&-1\\2&1&b\end{array}\right)[/tex]

list conditions on a and b such that there is:

i) no solution
ii) infinitely many solutions and
iii) a unique solution

The Attempt at a Solution



I row-reduced the matrix to

[tex]\left(\begin{array}{ccc}1&0&\frac{b+1}{2-a}\\0&1&\frac{-2-a}{2-a}\end{array}\right)[/tex]

and ended up with

i) for no solution, a = 0, 2 (since on the steps to row reducing, there was a [tex]\frac{1}{a}[/tex] in one of the entries). My textbook, however, says that a = 2, b != -1 are the conditions for no solution. Why b != -1 and why not a = 0?

Now I didn't get a result for unique solution and infinitely many solutions separately.

I only got that [tex]x1 = \frac{b+1}{2-a}[/tex] and [tex]x2 = \frac{-2 - ab}{2-a}[/tex]
which is the correct unique solution except that the book places restrictions on a and b, namely that they must be a = 2, b = -1. Where are these restrictions coming from and wouldn't a = 2 make it have no solution? Also how would one get an infinite amount of solutions in this case? Thanks.
 
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  • #2
Look at the result for x1 and x2. If a=2 there is no solution if either numerator doesn't vanish. But if they both have the form 0/0 then you have an infinite number of solutions. To verify this in a clearer way just put a=2 and b=-1 into the original equation. Then the first row and second rows of the matrix are the same, so you really only have one equation in two unknowns.
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in terms of vectors and matrices. It is used to solve problems involving systems of equations, transformations, and geometric interpretations.

2. What are linear equations?

Linear equations are mathematical expressions that contain variables raised to the first power and have a constant term. They can be expressed in the form of y = mx + b, where m is the slope and b is the y-intercept. In linear algebra, these equations can be represented as systems of equations and solved using various methods.

3. What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that are solved simultaneously to find values for the variables that satisfy all of the equations. The solutions to these equations can be represented as a point or points in n-dimensional space, where n is the number of variables in the system.

4. How do you solve a system of linear equations?

There are several methods for solving a system of linear equations, including substitution, elimination, and matrix operations. These methods involve manipulating the equations algebraically to eliminate one variable and solve for the others. Another approach is to use technology, such as graphing calculators or software, to solve the system numerically.

5. What are some applications of linear algebra?

Linear algebra has many applications in various fields, including physics, engineering, economics, and computer science. It is used to solve problems involving systems of equations, optimization, data analysis, and image processing. Some specific applications include predicting stock market trends, designing computer graphics, and analyzing network traffic.

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