Matrices and Systems of Equations

Take notes and use them to answer the questions.

Multiple Choice

A matrix derived from a system of linear equations is the _____ _____ of the system.

• augmented matrix
• constant matrix
• linear matrix
• coefficient matrix

Multiple Choice

Write the sytems of equations as an augmented matrix.

Multiple Choice

Write the system of linear equations for the augmented matrix.

Multiple Choice

Which choice is not one of the three elementary row operations described in the image?

• Interchange two rows

• Multiply a row by a nonzero constant

• Add a multiple of a row to another row

• Add a nonzero constant to a row

Multiple Choice

Which of the following is NOT an elementary row operation?

• Interchanging rows

• Adding a multiple of one row to another

• Multiplying a row by zero

• Multiplying a row by a non-zero constant

Multiple Choice

Perform the matrix row operation: $$0\cdot R_2$$

• $$0\ \ -8\ \ -6\ \ \ 16$$

• $$2\ \ -8\ \ -6\ \ \ 16$$

• $$0\ \ \ 0\ \ \ 0\ \ \ 16$$

• This is not a valid row operation!

Multiple Choice

Perform the matrix row operation: $$-1R_1$$ to create a new R₁

• $$1\ \ -2\ \ -8\ \ -7$$

• $$-2\ \ \ 1\ \ \ 7\ \ \ 6$$

• $$-2\ \ -2\ \ -8\ \ -7$$

• This is not a valid row operation!

Multiple Choice

Perform the matrix row operation: $$R_3-3R_1$$ to create a new R₁

• $$0\ \ \ 12\ \ -36\ \ -12$$

• $$-4\ \ \ 20\ \ \ 4\ \ \ 16$$

• $$-6\ \ \ 24\ \ \ 12\ \ \ 30$$

• This is not a valid row operation!

Multiple Select

What are the properties of a matrix in row-echelon form?

• Rows of zeros at the bottom

• Leading 1s in every row

• Leading 1s must be to the left

• All entries must be non-zero

Multiple Choice

Which rule is not correct for Gaussian elimination?

• Multiple rows by constant.

• Add/Subtract rows with rows.

• Divide by a constant

• Add/Subtract a number to a row.

• Swap rows

Multiple Choice

What does the first row look like in the augmented matrix?

• 7x + 5y     3
• 7     5     3
• 7x   +   5y   = 3
• none of these

Multiple Choice

What will be the matrix size for this system?

• 3x3

• 3x3

• 4x3

• 3x4

Multiple Choice

This is an example of

• Row Echelon Form

• Reduced Row Echelon Form

• Really Reduced Echelon Form

• Really Really Easy Form

Multiple Choice

Which row command is an appropriate next step?

• 7R2 + R3 -> R3

• -7R1 + R3 -> R3

• 7R3 -> R3

• -7R2 + R3 -> R3

Multiple Choice

What is the value of z?

• 6

• -6

• -36

• 4

​Things to remember about Augmented Matrices and Row Echelon Form.

If an equation in your system of equations is missing a variable. You will need to put a 0 coefficient for that variable in your augmented matrix.

A system of equations has infinite solutions if, when in row-echelon form, one entire row of your augmented matrix is zeros.

A system of equations has no solution when, in row-echelon form, the bottom row is all zeros except for the right-most element (element representing the constant).

Multiple Choice

When an element is missing in an equation, what value must go into the matrix for it?

• 1

• -1

• 0

• Nothing

Multiple Choice

If a matrix is in reduced row echelon form, then it is also in row echelon form.

• True

• False

Multiple Choice

Reduced Row Echelon Form is where ______________________

• Zeroes in the first column of my matrix

• Thel ast column has all zeroes

• One's are in a diagonal pattern of my matrix with zeroes underneath the one's.before the augmented portion

• One's are in a diagonal pattern of my matrix with zeroes above and below the ones before the augmented portion

Multiple Choice

Which of these matrices are in row echelon form?

• (a) only

• (b) only

• (a) and (d)

• (a) , (b), and (d)

Multiple Choice

Which of the following is in Row Echelon Form?

• A

• B

• C

• D

Multiple Choice

Jessica is simplifying the matrix using Gaussian Elimination. Did she complete the correct step?

• No, she should have changed the 2 to a zero and multiplied -2 by R2 and added R1

• No, she wanted to change the 3 to a zero so she should have multiplied R2 and added it to R3

• Yes. When you need a zero you multiply by the number's reciprocal.

• No. Just punch in the calculator. Who cares about Carl Gauss

Multiple Choice

The matrix is in row echelon form.
If  $$\left(x,y,z\right)$$  is the solution to the system represented by the matrix, which of the following is true?

• $$x=-3$$  

• $$y=-39$$  

• $$z=5$$  

• $$x=-16$$  

Multiple Choice

Write the augmented matrix in row echelon form.

Multiple Choice

The matrix represents a system of equations that has...

• infinitely many solutions

• one solution: (-10, -3, 1)

• no solution

• none of the above

Multiple Choice

The augmented matrix shown in row-echelon form indicates a system with

• one solution

• no solution

• infinitely many solutions

Multiple Choice

What is the condition for a system of linear equations represented by AX = B to have a unique solution?

• A must be a singular matrix

• A must be an invertible matrix

• B must be a zero matrix

• X must be a non-zero matrix

​Solving a System Using an Inverse Matrix

Find the inverse of the coefficient Matrix.

​Multiply the coefficient matrix by the constant matrix

check: (4) +3(-1) = 1 and 2(4) +5(-1) = 3

Multiple Choice

What is the matrix equation to solve this system of equations?

Multiple Choice

To solve this system of equations by using $$x\ =\ A^{-1}b$$ , you need $$A^{-1}$$ . Which matrix is $$A^{-1}$$ ?

Fill in the Blank

Solve the system by inverting the coefficient matrix and using (A^-1)(b)=x. Enter your answer as (x1, x2) format.

(

,

-

)

Multiple Choice

Solve the system

• x=-3

  y=4

• x=-2

  y=5

• x=3

  y=-4

• x=2

  y=-5

Multiple Choice

Solve the system.

• x=-1

  y=2

  z=-2

• x=2

  y=1

  z=-1

• x=-2

  y=-1

  z=-2

• x=2

  y=2

  z=-1

Multiple Choice

Lesson Review: What is the augmented matrix of a system of equations?

• The augmented matrix is a matrix where each element represents the solution to the system of equations.

• The augmented matrix of a system of equations is a matrix where each row represents one equation, with the coefficients of the variables followed by the constant term.

• The augmented matrix is a matrix where each column represents one equation.

• The augmented matrix is a matrix where each row represents one variable.

Multiple Choice

Lesson Review: How can matrices be used to represent a system of linear equations?

• By randomly assigning values to the elements of the matrices

• By setting up a matrix equation Ax = B, where A is the coefficient matrix, x is the variable matrix, and B is the constant matrix.

• By using matrices to represent non-linear equations

• By ignoring the constants in the system of linear equations