Linear Algebra/Topic: Cramer's Rule/Solutions

SolutionsEdit

Problem 1

Use Cramer's Rule to solve each for each of the variables.

  1.  
  2.  
Answer
  1.  ,  
  2.  ,  
Problem 2

Use Cramer's Rule to solve this system for  .

 
Answer

 

Problem 3

Prove Cramer's Rule.

Answer

Determinants are unchanged by pivots, including column pivots, so   is equal to   (use the operation of taking   times the first column and adding it to the  -th column, etc.). That is equal to  , as required.

Problem 4

Suppose that a linear system has as many equations as unknowns, that all of its coefficients and constants are integers, and that its matrix of coefficients has determinant  . Prove that the entries in the solution are all integers. (Remark. This is often used to invent linear systems for exercises. If an instructor makes the linear system with this property then the solution is not some disagreeable fraction.)

Answer

Because the determinant of   is nonzero, Cramer's Rule applies and shows that  . Since   is a matrix of integers, its determinant is an integer.

Problem 5

Use Cramer's Rule to give a formula for the solution of a two equations/two unknowns linear system.

Answer

The solution of

 

is

 

provided of course that the denominators are not zero.

Problem 6

Can Cramer's Rule tell the difference between a system with no solutions and one with infinitely many?

Answer

Of course, singular systems have   equal to zero, but the infinitely many solutions case is characterized by the fact that all of the   are zero as well.

Problem 7

The first picture in this Topic (the one that doesn't use determinants) shows a unique solution case. Produce a similar picture for the case of infintely many solutions, and the case of no solutions.

Answer

We can consider the two nonsingular cases together with this system

 

where   of course yields infinitely many solutions, and any other value for   yields no solutions. The corresponding vector equation

 

gives a picture of two overlapping vectors. Both lie on the line  . In the   case the vector on the right side also lies on the line   but in any other case it does not.