Solving Systems with Augmented Matrices

Solving Systems with Augmented Matrices

​But first let's review

​Solving systems using Matrix Inverse

Multiple Choice

Which numbers are correct for the first row in the coefficient matrix?

• 5 0 -10

• 2 3 -1

• -6 1 -4

• 5 0 0

Multiple Choice

Which numbers are correct for the 2nd row in the coefficient matrix?

• 2 3 0

• 2 3 -1

• -6 1 -4

• 5 0 0

Multiple Choice

Which numbers are correct for the 3rd row in the coefficient matrix?

• 2 3 0

• -6 1 4

• -6 1 -4

• -6 0 4

Multiple Choice

Which numbers are correct for the constant matrix?

• -10

  -1

  -11

• -10 -1 -11

• 10

  1

  11

• 10 1 11

Multiple Choice

When solving system using the inverse, which two matrices do you multiply?

• coefficient matrix is multiplied by the constant matrix

• the inverse of the coefficient matrix is multiplied by the constant matrix

• coefficient matrix is multiplied by the solution matrix

• the inverse of the coefficient matrix is multiplied by the solution matrix

Multiple Choice

Solve for x, y and z.

• (2, 1, 6)
• (1, -2, 6)
• (6, 1, 2)
• (-2, 1, 6)

Multiple Choice

Solve the system of linear equations using the inverse matrix.

• (3, 1, -5)

• (-3, 1, -1)

• not possible; the inverse does not exist

Multiple Choice

Solve the system of equations using Inverse Matrix: 

• ( -2, 4, 3)
• ( 2, 1, 6)
• Not possible; the inverse does not exist.

Open Ended

When should you use augmented matrices?

Fill in the Blanks

What does rref stand for?

Multiple Choice

What are you to look for in the solution matrix to know the solution is unique, an ordered pair or triple?

• the first part of the solution matrix will look like the identity matrix

• The solution matrix will be a square matrix

• One of the rows in the solution matrix will have all zeros

• One of the rows in the solution matrix will have zeros and 1 number.

Multiple Choice

What in the answer matrix indicates that there is no solution?

• the first part of the solution matrix will look like the identity matrix

• The solution matrix will be a square matrix

• One of the rows in the solution matrix will have all zeros

• One of the rows in the solution matrix will have zeros and 1 number.

Multiple Choice

What in the answer matrix indicates that there is an infinite number of solutions, infinitely many solutions?

• the first part of the solution matrix will look like the identity matrix

• The solution matrix will be a square matrix

• One of the rows in the solution matrix will have all zeros

• One of the rows in the solution matrix will have zeros and 1 number.

​Compare the three possible solutions on the next slide. Pay close attention to the zeros.

Now let's learn how to enter the augmented matrices into the calculator.​

Multiple Choice

Where is the rref function found on the calculator?

• In the matrix menu

• In the function window

• In the mode function

• in the math menu

Multiple Select

What two keystrokes get you into the matrix menu?

• 2nd

• x^-1

• alpha

• zoom

Multiple Select

What keystroke will move you to the Math list in the matrix menu?

• arrow up

• arrow right

• arrow down

• arrow left

Multiple Select

Where is rref found within the Math list in the matrix menu?

• A:

• B:

• 1:

• 8:

Multiple Choice

Which direction must you scroll to find rref?

• up

• down

• doesn't matter, I can find rref moving in either direction

Multiple Select

What two keystrokes will pull up the box for you to type in the dimensions of the augmented matrix?

• 2nd

• x^-1

• alpha

• zoom

Multiple Choice

The coefficients and the constants are all included in an augmented matrix.

• yes

• no

Multiple Choice

What are you to look for in the solution matrix to know the solution is unique, an ordered pair or triple?

• the first part of the solution matrix will look like the identity matrix

• The solution matrix will be a square matrix

• One of the rows in the solution matrix will have all zeros

• One of the rows in the solution matrix will have zeros and 1 number.

Multiple Choice

What in the answer matrix indicates that there is an infinite number of solutions, infinitely many solutions?

• the first part of the solution matrix will look like the identity matrix

• The solution matrix will be a square matrix

• One of the rows in the solution matrix will have all zeros

• One of the rows in the solution matrix will have zeros and 1 number.

Multiple Choice

What in the answer matrix indicates that there is no solution?

• the first part of the solution matrix will look like the identity matrix

• The solution matrix will be a square matrix

• One of the rows in the solution matrix will have all zeros

• One of the rows in the solution matrix will have zeros and 1 number.

​Now let's do those examples again but this time we'll use our new method.

Multiple Choice

What are the numbers in the first row of the augmented matrix?

• 1 1 1 -3

• 1 -1 1

• 1 -1 1 -3

• 1 1 1 3

Multiple Choice

What are the numbers in the second row of the augmented matrix?

• 1 -1 -1 -3

• 1 -1 -1

• 1 -1 1 -3

• 1 1 1 -3

Multiple Choice

What are the numbers in the 3rd row of the augmented matrix?

• 5 5 1 -15

• 5 -5 1 -15

• 5 -5 0 -15

• 5 5 1 15

Multiple Select

Which row is included in the solution matrix?

• 1 -1 0 -3

• 0 0 1 0

• 0 0 0 0

Multiple Choice

Solve the system of equations using rref (augmented matrices): 

• Unique Solution:

  ( 2, 1, 6)

• infinitely many solutions

• No solution

Multiple Choice

Solve the system of linear equations.

• (3, 1, -5)

• (-3, 1, -1)

• infinitely many solutions

• no solution

Multiple Choice

Solve the system of equations: 

• ( -2, 4, 3)
• ( 2, 1, 6)
• Infinitely Many Solutions

• No solution

Multiple Choice

Solve the system:

-4x - 5y - z = 18

-2x - 5y - 2z = 12

-2x + 5y + 2z = 4

• (4, 0, -2)
• (-4, 0, -2)
• Many Solutions
• No Solution

Fill in the Blanks

$5x–4y+z=-18$    $x+3y+7z=38$  

$4x+2y–6z=-38$  

Multiple Choice

$$4x+2y-6z=34$$ $$2x+y+3z=3$$

$$6x+3y–3z=37$$

• (5,-7/3,0)
  

• No Solution

• Infinite Solution

• (-5,7/3,0)

Multiple Choice

$$-3x+3y+6z=4$$ $$3x+y+2z=3$$

$$-6x+2y+4z=34$$

• (5,-7/3,0)
  

• No Solution

• Infinite Solution

• (-5,7/3,0)