Solving a system using Cramer's rule or matrix equation

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masters

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Here's the system:

2a + b - c = -6
a - 2b + c = 8
-a - 3b + 2c = 14

The book says the answer is (0, -2, 4). However, the determinant of the coefficient matrix is 0. The matrix is singular and has no inverse, so Cramer's rule has you dividing by 0, and since it has no inverse, you can't use a matrix equation to solve. The given solution is a solution. My question is whether it's the only solution.
 
When Cramer's Rule won't work, this means that the system is either inconsistent ("no solution") or dependent ("many solutions"). In such a case, one must do the row operations to reduce the system to something that may be interpreted algebraically. In other words, one much resort to earlier, messier methods. :wink:

In this case, since the book gives one specific solution, the system must be independent. So you might want to check your value for the determinant. It is not zero! :oops:

Eliz.
 
masters said:
The book says the answer is (0, -2, 4). However, the determinant of the coefficient matrix is 0.
Click to expand...
That is correct. Thus there might be a typo or else the book's answer is just wrong.

If we change -a - 3b + 2c = 14 to a - 3b + 2c = 14 then the book's answer is correct.
 
masters said:
2a + b - c = -6
a - 2b + c = 8
-a - 3b + 2c = 14

The book says the answer is (0, -2, 4).
Click to expand...
pka said:
If we change -a - 3b + 2c = 14 to a - 3b + 2c = 14 then the book's answer is correct.
Click to expand...
Checking the solution in the original system, I get the following:

. . . . .2[0] + [-2] - [4] = 0 - 2 - 4 = -6

. . . . .[0] - 2[-2] + [4] = 0 + 4 + 4 = 8

. . . . .-[0] - 3[-2] + 2[4] = 0 + 6 + 8 = 14

So it looks, to me at least, as though the profferred solution "works" in the posted system of equations. But I'd miscalculated the determinant earlier. :oops:

Code: 
| 2  1 -1 |
| 1 -2  1 | = 2|-2 1 | - 1| 1  1 | + (-1)| 1 -2 |
|-1 -3  2 |    |-3 2 |    |-1  2 |       |-1 -3 |

            = 2[(-2)(2) - (-3)(1)] - 1[(1)(2) - (-1)(1)] - 1[(1)(-3) - (-1)(-2)]

            = 2[-4 + 3] - 1[2 + 1] - 1[-3 - 2]

            = 2[-1] - 1[3] - 1[-5] = -2 - 3 + 5 = 0
The full solution, from the row-reduced matrix:

Code: 
[1  0  -1/5   -4/5]
[0  1  -3/5  -22/5]
[0  0    0     0  ]
...is (a, b, c) = ((1/5)c - 4/5, (3/5)c - 22/5, c). That is, the system is dependent.

Eliz.
 
Thanks, Stapel. I have found quite a few wrong answers stated in the Glencoe Alg 2 2005 edition. Have a great holiday season.
 
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