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Examples of Cramer’s Rule
Use Crammer’s Rule to solve:
3x1 + 4x2 − 3x3 = 5
3x1 − 2x2 + 4x3 = 7
3x1 + 2x2 − x3 = 3.
In matrix form Ax = b or a1 a2 a3 x = b this is


  
3 4 −3
x1
5
 3 −2 4   x2  =  7  .
3 2 −1
x3
3
b a2 a3
Cramer’s Rule says that x1 =
5 4 −3
7 −2 4
3 2 −1
x1 = 3 4 −3
3 −2 4
3 2 −1
a1 b a3
, x3 =
3 5 −3 3 4
3 7 4 3 −2
3 3 −1 3 2
, x2 = , x3 = 3 4 −3 3 4
3 −2 4 3 −2
3 2 −1 3 2
|A|
, x2 =
|A|
a1 a2 b
|A|
. So
5 7 3 .
−3 4 −1 Expanding, we get:
3 4 −3 4 −3 4 −3 −2 4 3 −2 4 = 3 2 −1 − 3 2 −1 + 3 −2 4 = 3(−6) − 3 · 2 + 3 · 10 = 6.
3 2 −1 5 4 −3 4 −3 4 −3 −2 4 7 −2 4 = 5 2 −1 − 7 2 −1 + 3 −2 4 = 5(−6) − 7 · 2 + 3 · 10 = −14.
3 2 −1 3 5 −3 3 7 4 = 3 7 4 − 3 5 −3 + 3 5 −3 = 3(−19) − 3 · (−4) + 3 · 41 = 54.
3 −1 3 −1 7 4 3 3 −1 3 4 5 3 −2 7 = 3 −2 7 − 3 4 5 + 3 4 5 = 3(−20) − 3 · 2 + 3 · 38 = 48.
2 3 2 3 −2 7 3 2 3 x1 =
−14
7
54
48
= − , x2 =
= 9, x3 =
= 8.
6
3
6
6
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Review for Midterm I
Topics:
1. Solve systems of linear equations using
(a) Gaussian Elimination and back substitution (echelon form)
(b) Gauss-Jordan Elimination (reduced echelon form)
2. For a system of linear equations whose matrix is in row echelon form
(a) Determine how many solutions it has
(b) Describe (parameterize) its solutions if there are infinitely many
3. Do matrix arithmetic including multiplication by constants, addition, multipication
and solving equations
4. Find the inverse of a matrix (especially a 3 × 3 matrix) using
(a) Row transformations
(b) Cramer’s Rule, i.e., adjoint formula
5. Use the "inverse formula" X = A−1 B for solving a system of linear equations AX = B
6. Calculate determinants using
(a) Expansion by Minors
(b) Gaussian Elimination
7. Use Cramer’s Rule to solve systems of linear equations, especially 2 × 2 systems.
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Sample Test
Instructions: No calculators; show your work NEATLY.
Comment: “Hints for checking” will not appear on the real test.
1. [15 points] How many solutions does each of the following linear systems have? Circle
the entries which led to your conclusion.


1 0 0 0 −2
 0 1 0 0 2 


−1 
0
0
1
0
(a) 


 0 0 0 1 −2 
0 0 0 0 0


1 0 1 −1
(b)  0 1 −3 −3 
0 0 0 −1


1 0 0 1 0 −6
 0 1 −5 1 −1 −5 

(c) 
 0 0 0 0 0 0 
0 0 0 0 0 0
2. [15 points] Use Cramer’s rule to solve the following system of linear equations for x.
−x − 2y = 1
−3x + y = 0
[Hint for checking: Also solve for y and plug your answer back into the equations.]
3. [15 points] Parameterize the solutions to the linear system whose matrix is below.
Assume the original variables were x1 , x2 , x3 , x4 , x5 .


1 −3 0 −2 0 11
 0 0 1 −1 0 −1 


 0 0 0 0 1 4 .
0 0 0 0 0 0
[Hint for checking: Plug your answer back in to the original equations. If there are free
variables, first set them to 0 then to 1.]
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4. [15 points] Solve the linear system
−11x1 + 16x2 + 10x3 + 6x4
−4x1 + 3x2 + 2x3 + 2x4
4x1 − 5x2 − 3x3 − 2x4
−2x1 + 3x2 + 2x3 + x4
=
=
=
=
84
18
−26
16
using the fact that
−1 
−11 16 10 6
1 0
2 −2
 −4 3

 0 −1 −2 −2
2
2



 4 −5 −3 −2  =  0 1
3
4
−2 3
2
1
2 1
4 −5



.

Other methods will result in the loss of points. [Hint for checking: Plug your answer
into the original equation.]
−3 2
2
3
0
0 −2 0 5. [20 points] Find . [Hint for checking: Use more than one method—
0 −2 −2 0 3
0
2 4 row transformations and expansion by minors.]


1 0 1
6. [20 points] Find the inverse of  1 −1 0  using Gauss-Jordan Elimination. [Hint
5 −4 0
for checking: Do matrix multiplication to check that AA−1 = I.]
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