Solving Systems of Equations by matrices Chapter 4.4 Solving Systems of Equations by matrices

Objectives Use matrices to solve a system of two equations. Use matrices to solve a system of three equations.

What is a matrix? A matrix (plural: matrices) is a rectangular array of numbers. The following are examples of matrices:

Matrices The numbers aligned horizontally in a matrix are in the same row. The numbers aligned vertically are in the same column. This matrix has 2 rows and 3 columns. It is called a 2 X 3 (read “two by three”) matrix. row 1 row 2 column 1 column 2 column 3

How do equations and matrices relate? System of Equations Corresponding (in standard form) Matrix The rows of the matrix correspond to the equations in the system. The Coefficients of each variable are placed to the left of the vertical dashed line. Matrices are a shorthand notation for representing systems of equations.

Elementary Row Operations Any two rows in a matrix may be interchanged. The elements of any row may be multiplied (or divided) by the same nonzero number. The elements of any row may be multiplied (or divided) by a nonzero number and added to their corresponding elements in any other row. Notice that these row operations are the same operations that we can perform on equations in a system.

Solving a system of 2 equations by matrices Matrix System of Equations In the second equation, we have y = 5. Substituting this in the first equation, we have x + 2(5) = -3 or x = - 13. The solution of the system in ordered pair is ( -13, 5)

Example 1: Use matrices to solve the system: Step 1: Set up the corresponding matrix

Example 1: Continued Step 2: Use elementary row operations to write an equivalent matrix that looks like: In the given matrix, the element in the first row is already 1, as desired. Next we write an equivalent matrix with a 0 below the 1. To do this, we multiply row 1 by -2 and add to row 2. We will change only row 2. Simplifies to: Row 2 element Row 1 element Row 2 element Row 1 element Row 2 element Row 1 element

Example 1: Continued Step 3: Now we change -7 to a 1 by use of an elementary row operation. We dived row 2 by -7

Example 1: Continued The last matrix corresponds to the system: To find x, we let y = 2 in the first equation. x + 3(2) = 5 x = -1 The ordered pair solution is (-1, 2). Check to see that this ordered pair satisfies the equations.

Use matrices to solve: Step 1: Set up matrix Let’s try another one!  Use matrices to solve: Step 1: Set up matrix Step 2: Write an equivalent matrix that looks like: To do this, multiply the first row by -2 and add to row 2. Leave row 1 the same! Simply take the coefficient from each variable…

Step 4: Write the system that corresponds to the matrix: Let’s try another one!  Step 3: Now change the -7 to 1 by using elementary row operation. We divide row 2 by -7 Step 4: Write the system that corresponds to the matrix: Step 5: Substitute to find the unknown variable. x + 2y = -4 x + 2 (-3) = -4 x – 6 = -4 x = 2 The ordered pair solution is (2, -3)

Give it a try!  Use matrices to solve: Solution: (2, -1)

Example 2: Use matrices to solve the system: Step 1: Set up a corresponding matrix Step 2: To get 1 in the row 1, column 1 position, we divide the elements of row 1 by 2.

Example 2 Continued Step 3: To get 0 under 1, we multiply the elements of row 1 by -4 and add the new elements to the elements of row 2. Step 4: Write a corresponding system: The equation 0 = -1 is false for all y or x values; hence the system has no solution!

Concept Check: Consider the system What is wrong with the corresponding matrix shown below? Answer bottom left on page 251

Give it a try!  Use matrices to solve: Solution: Null Set

Solving a system of three equation in three variables using matrices The matrix must be written in the following form:

Example 3 Use matrices to solve the system: Step 1: Set up a corresponding matrix Our goal is to write an equivalent matrix with 1’s along the diagonal…see the numbers in red and 0’s below the 1’s. The element in row 1, column 1 is already 1. Next we get 0’s for each element in the rest of column 1. The numbers in blue need to get changed to zeros.

Example 3: Continued Step 2: Multiply the elements in row 1 by 2 and add the new elements to row 2. Multiply the elements of row 1 by -1 and add the new elements to the elements of row 3. We do not change row 1! Green = Row 1 Pink = # multiplied by

Example 3 Continued Step 4: We continue down the diagonal and use elementary row operations to get 1 where the element 3 is now. To do this, we interchange rows 2 and 3. is equivalent to

Example 3: Continued Step 5: Next we want the new row 3, column 2 element to be 0. We multiply the elements of row 2 by -3 and add the result to the elements of row 3. Green = Row 2 Pink = # multiplied by

Example 3: Continued Step 6: Finally, we divide the elements of row 3 by 13 so that the final diagonal element is 1. Step 7: Write the system that corresponds to the matrix. Step 8: Substitute z = 3 into the 2nd equation y – 3(3) = -10 y – 9 = -10 y = -1 Step 9: Substitute z =3 and y = -1 into the 1st equation x + 2(-1) + 3 = 2 x – 2 + 3 = 2 x + 1 = 2 x = 1 Ordered triple solution: (1, -1, 3)

Give it a try!  Use matrices to solve: Ordered triple solution: (1, 2, -2)