Course 111: Algebra, 2nd Feb 2007
To be handed in at tutorials on Feb 5th and 6th.
1. Solve the following system of linear equations using both Gaussian
elimination and Gauss-Jordan elimination.
3x1 − 4x2 = 10
−5x1 + 8x2 = −17
−3x1 + 12x2 = −12
What is the rank of the coefficient matrix describing this system?
2. Solve the following system of linear equations using both Gaussian
elimination and Gauss-Jordan elimination.
7x1 + 2x2 − 2x3 − 4x4 + 3x5 = 8
−3x1 − 3x2 + 2x4 + x5 = −1
4x1 − x2 − 8x3 + 20x5 = 1
What is the rank of the coefficient matrix describing this system?
3. Prove that if A is an n × n matrix (ie. A describes a system with as
many equations as unknowns) then the reduced row-echelon form of
the matrix will either contain at least one row of all zeroes or it will be
the n × n identity matrix, In where


1 ...
 . .

I =  .. . .
0 ...
0
.. 
. 

{z
}
|
1
n cols, n rows