http://www.ck12.org Chapter 8. Systems and Matrices3.
[ 3 4 1 0
5 −1 0 1]→ · 5 →
→ · 3 →
[15 20 5 0
15 −3 0 3
] →
→ −I →
[ 15 20 5 0
0 − 23 −5 3
]
→ · 23 →
→ · 20 →
[ 345 460 115 0
0 − 460 −115 60
]
[ 345 460 115 0
0 − 460 −115 60
]→ +II →
→
[ 345 0 0 60
0 − 460 −115 60
]
→ ÷ 345 →
→ ÷− 460 →
[1 0 0 60
345
0 1^115460 − 46060]
Notice how fractions were avoided until the final step. Adding and subtracting large numbers in a matrix is easier
to handle than adding and subtracting small numbers because then you don’t need to find a common denominator.Practice- Give an example of a matrix in row echelon form.
- Give an example of a matrix in reduced row echelon form.
- What are the three row operations you are allowed to perform when reducing a matrix?
- If a square matrix reduces to the identity matrix, what does that mean about the rows of the original matrix?
Use the following matrix for 5-6.
A=
− 3 − 4 − 12
4 4 12
− 11 − 12 − 35
- Reduce matrixAto row echelon form.
- Reduce matrixAto reduced row echelon form. Are the rows of matrixAlinearly independent?
Use the following matrix for 7-8.
B=
3 − 4 8
9 0 1
0 1 − 2
- Reduce matrixBto row echelon form.
- Reduce matrixBto reduced row echelon form. Are the rows of matrixBlinearly independent?
Use the following matrix for 9-10.
C=
0 0 − 1 − 1
3 6 − 3 1
6 12 − 7 0
- Reduce matrixCto row echelon form.
- Reduce matrixCto reduced row echelon form. Are the rows of matrixClinearly independent?
Use the following matrix for 11-12.
D=
1 1
3 4
2 3
- Reduce matrixDto row echelon form.