http://www.ck12.org Chapter 8. Systems and Matrices
y=
∣∣
∣∣∣
∣∣
∣a j c
d k f
g l i∣∣
∣∣∣
∣∣
∣ ∣
∣∣∣
∣∣
∣∣a b c
d e f
g h i∣∣
∣∣∣
∣∣
∣=
∣∣
∣∣∣
∣∣
∣1 0 − 1
7 14 1
0 10 1
∣∣
∣∣∣
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∣ ∣
∣∣∣
∣∣
∣∣1 2 − 1
7 0 1
0 1 1
∣∣
∣∣∣
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∣=^140 ++^00 +(+(−−^707 )−)− 00 −− 110 − 14 −^0 =−−^6622 = 3
Concept Problem Revisited
Example C reminds you of the fact that a problem done with traditional coefficient elimination can take over a page
of writing and rewriting. Efficiency partly means requiring less time and space. If this was all that efficiency
meant then it would not make sense to solve systems of two equations with two unknowns using matrices because
the solution could be found more quickly using substitution. However, the other part of efficiency is minimizing
the number of decisions that have to be made. A computer is very good at adding, subtracting and multiplying
numbers, but not very good at deciding whether eliminatingxor eliminatingywould be better. This is why a
definite algorithm using matrices and Cramer’s Rule is more efficient.
Vocabulary
Amatrix equationrepresents a system of equations by multiplying a coefficient matrix and a variable matrix to get
a solution matrix.
Guided Practice
- Solve the following system using Cramer’s Rule.
5 x+ 12 y= 72
18 x− 12 y= 108- Solve the following system using Cramer’s Rule and your calculator.
70 x+ 21 y=− 112
27 x− 21 y= 15- What is the value ofzin the following system?
3 x+ 2 y+z= 7
4 x+ 0 y+z= 6
6 x−y+ 0 z= 5Answers: