Instructions. Answer the five questions (20 points each) below to the best of your knowledge. Must show work to receive partial credit. Good Luck

Matrices (Question 1-5)

1. Which of the following can you not do when solving a system of equations using matrices?
   1. Add two rows.
   2. Switch two rows.
   3. Add a constant to a row.

Note: When performing row operations on a matrix, we can:

Add two rows together.

Switch two rows.

Multiply a row by a constant.

These operations are known as elementary row operations and are essential for transforming matrices into row-echelon form or reduced row-echelon form, which are used to solve systems of linear equations.

1. Multiply a row by a constant.

1. When representing a system of linear equations in two variables as a matrix equation, one can use the general form A[xy]=b A[xy]=b. For a given system of linear equations, is it possible for the 2-by-2 matrix AA to have different entries? Why or why not?

1. Yes. Either equation in the system of equations can be multiplied by a constant. This will also affect b b.
2. No. A different matrix AA would result in a different answer regardless of anything else.

Note:In a system of linear equations, the matrix A represents the coefficients of the variables. If the entries of matrix A change, the relationships defined by the equations also change, leading to a different solution set. Thus, different entries in matrix A imply a different system of equations.

1. It depends on whether the system of equations is dependent or independent.
2. No. Only a system of equations that is inconsistent, and therefore has no answer anyways, can have different AA matrices.

1. Jake is going to solve the system of equations using a matrix equation. He sets up his matrix equation as follows. [5621][xy]=[37][5-26-1][xy]=[37] Is this a, correct? If not, choose the correct reason why not.

1. It is correct.
2. There must be a fraction in the matrix equation since there is one in the system of equations.
3. The coefficients don't match up to the correct variables in the matrix equation.

Note: This option is correct if the coefficients do not align properly with the variables. The matrix equation should accurately reflect the coefficients of the variables in the system of equations. If the coefficients are mismatched, the matrix equation will not correctly represent the system of equations.

1. Systems of linear equations with equations not in standard form can never be put into a matrix equation.

1. The first element of the first row will be 2a+52a+5 and the first element of the second row will be 4+5b-4+5b. Since these have to be equal to the elements of the matrix on the right-hand side, this means that 2a+5=132a+5=13 and 4+5b=9-4+5b=-9. Solving these equations yields a=4a=4 and b=1b=-1.

Is Simon's work, correct? If not, where did he make a mistake?

1. This is correct.
2. This is not correct. Simon cannot move the parenthesis as he did in his first step.
3. This is not correct. Simon made a mistake in multiplying the two matrices.
4. This is not correct. Simon only did part of the matrix multiplication to find the two equations. The full matrix multiplication will yield four equations, leading to an overdetermined system of equations that is inconsistent.

Note: This option indicates that Simon did not complete the multiplication process. In matrix multiplication, each element of the resulting matrix is derived from the sum of the products of the corresponding elements of the rows and columns of the matrices being multiplied. If Simon only considered part of the multiplication, he would not have accounted for all necessary equations, leading to an incomplete and potentially inconsistent system.

1. Two or more linear equations in the same variable form
2. a point of intersection.
3. a solution of a linear system.

Note: When you have two or more linear equations in the same variable, they form a system of linear equations.A system of linear equations can have one solution, infinitely many solutions, or no solution at all, depending on the equations.

1. perpendicular lines.
2. a system of linear equations.