RESPOND TO DISCUSSION QUESTIONS AND RESPONSES ACCORDINGLY. NUMBER RESPONSES ACCORDINGLY.
TJ1. Under what conditions will a matrix have an inverse such that their product is the identity matrix?
Requirements to have an Inverse
- The matrix must be square (same number of rows and columns).
- The determinant of the matrix must not be zero (determinants are covered in section 6.4). This is instead of the real number not being zero to have an inverse, the determinant must not be zero to have an inverse.
A square matrix that has an inverse is called invertible or non-singular. A matrix that does not have an inverse is called singular.
A matrix does not have to have an inverse, but if it does, the inverse is unique.
SM2. Well, in real numbers, the inverse of any real number a was the number a-1, such that a times a-1 equaled 1. We knew that for a real number, the inverse of the number was the reciprocal of the number, as long as the number wasn’t zero.
The inverse of a square matrix A, denoted by A-1, is the matrix so that the product of A and A-1 is the Identity matrix. The identity matrix that results will be the same size as the matrix A. Wow, there’s a lot of similarities there between real numbers and matrices. That’s good, right – you don’t want it to be something completely different.
A(A-1) = I or A-1(A) = I
There are a couple of exceptions, though. First of all, A-1 does not mean 1/A. Remember, “There is no Matrix Division!” Secondly, A-1 does not mean take the reciprocal of every element in the matrix A.
EH3r. How is the determinant calculated?
EH4. Given any system of two linear equations in two unknowns, and using either the addition or the substitution method, how can you determine when the system is inconsistent or dependent?
EH5. Suppose you are given a system of linear equations. What are the advantages and disadvantages of using the substitution and elimination methods to solve the system?
DIS# 2
EH2 1. Can you look at the system of equations and without solving them, determine if they are consistent or inconsistent?
AF2 2. In general, a solution of a system in two variables is an ordered pair that makes BOTH equations true.
In other words, it is where the two graphs intersect, what they have in common. So if an ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the system.
A consistent system is a system that has at least one solution. An inconsistent system is a system that has no solution.
The equations of a system are dependent if ALL the solutions of one equation are also solutions of the other equation. In other words, they end up being the same line.
The equations of a system are independent if they do not share ALL solutions. They can have one point in common, just not all of them.
SM2 2. A system of equations that has at least one solution is Consistent
a system of equations that has no solution is inconsistent
If 2 equations has infinite solutions, it is dependent
If it has 1 or no solution, it is independent
So, graph it. If it has a solution set, it’s consistent and if the lines intersect in one place or not at all it’s independent. If it’s the same line, it’s dependent
16x-8y=40
-3x+8y=-14
13x = 26
x =2 y = –1 so it has a solution set and they don’t have the same slope, so they’re not the same line on the graph.