Augmented Matrices
Graphing Technology Lab TI-Nspire™ Augmented Matrices
Using TI-Nspire technology, you can solve a system of linear equations using the Matrix & Vector function. An augmented matrix contains the coefficient matrix with an extra column containing the constant terms. You can use a graphing calculator to reduce the augmented matrix so that the solution of the system of equations can be easily determined.
EXAMPLE
Write an augmented matrix for the following system of equations. Then solve the system by using a graphing calculator. 2x + y + z = 1 3x + 2y + 3z = 12 4x + y + 2z = -1 EXAMPLE 1 Step 1 Write the augmented matrix and enter it into a calculator. 2 1 1 1 The augmented matrix B = 3 2 3 12 . 4 1 2 -1 Begin by creating the augmented matrix from the matrix of coefficients and the matrix of constants.
KEYSTROKES: c 1: Calculator b 6: Matrix & Vector 6: Create 7: Augment
Press / r. Arrow until the is highlighted and then press ·. Then enter the dimensions of the matrix for the coefficients and enter those values.
KEYSTROKES: 3 e 3 e · 2 e 1 e 1 e 3 e 2 e 3 e 4 e 1 e 2 e ,
Press / r. Arrow until the is highlighted and then press · . Then enter the dimensions of the constants and enter those values.
KEYSTROKES: 3 e 1 e · 1 e 12 ev 1 e · EXAMPLE 1 The augmented matrix will appear on the right side of screen. Step 2 Find the reduced row echelon form using the graphing calculator.
KEYSTROKES : b 6: Matrix & Vector 4: Reduced Row-Echelon Form
Use the arrows to highlight the augmented matrix. Then press / C to copy it. Use the arrow keys until the cursor is between the parentheses and press / V to paste the copied matrix into place. Press ·. Study the reduced echelon matrix. The first three columns are the same as a 3 × 3 identity matrix. The first row represents x = -4, the second row represents y = 3, and the third row represents z = 6. The solution is (-4, 3, 6).
Nspire_236_C04L6B_888482.indd 236 8/14/09 3:00:21 PM Exercises Write an augmented matrix for each system of equations. Then solve with a graphing calculator. 1. 3x + 2y = -4 2. 2x + y = 6 3. 2x + 2y = -4 4x + 7y = 13 6x - 2y = 0 7x + 3y = 10 4. 4x + 6y = 0 5. 6x - 4y + 2z = -4 6. 5x - 5y + 5z = 10 8x - 2y = 7 2x - 2y + 6z = 10 5x - 5z = 5 2x + 2y + 2z = -2 5y + 10z = 0
Nspire_236_C04L6B_888482.indd 237 8/14/09 3:00:24 PM