3. (9 pts) (a) Explain the difference between a coefficient matrix and an augmen

Question

3. (9 pts) (a) Explain the difference between a coefficient matrix and an augmented matrix. (b) If you know that a system of linear equations has at least two solutions, how many solutions must it actually have? Explain (c) How many solutions can a consistent system of linear equations have? Discuss the different possibilities 4. (12 pts) (a) Give an example of two row equivalent matrices in echelon form. Note that although the reduced echelon form is unique, there will be several different row equivalent matrices in (b) Suppose a 4 x 6 coefficient matrix has 4 pivot columns. Is the corresponding system of (c) if a 7 × 5 augmented matrix has a pivot in every column, what can you say about the (d) If a consistent system of equations has more unknowns than equations, what can be said echelon form. equations consistent? Justify your answer solutions to the corresponding system of equations? Justify your answer about the number of solutions?

Explanation / Answer

3.

a)

Suppose you have a system of linear equations like

2x1 + 3x2 = 5

-x1 + 4x2 = 7

Then the coefficient matrix for this system is the matrix

and the augmented matrix for this system is the matrix

The coefficient matrix captures the information on the left side of the equals signs. The augmented matrix captures the information on both sides.

If we were to write this system as a matrix equation Ax=b, then A would be the coefficient matrix.

b)

Two lines will never intersect, have no solutions if they are parallel. Two lines will lay on top of one another, and have infinitely many solutions, if they have the same slope and intercept y at the same point. If the lines have different slopes, they will intercept at one point. The key is remembering that these are LINEAR equations, meaning straight lines, which limits the possibilities to one, none, or infinite. SO answer is it can have infinite solutions

c)

All the systems of equations that have unique solutions These are referred to as Consistent Systems.

A system of two linear equations can have one solution, an infinite number of solutions, or no solution. Systems of equations can be classified by the number of solutions.

If a system has at least one solution, it is said to be consistent .

If a consistent system has exactly one solution, it is independent .

If a consistent system has an infinite number of solutions, it is dependent . When you graph the equations, both equations represent the same line.

All The Best Dear have a nice Day

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