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Exam Cram Winter 2011
MATH 204 Handouts
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Exam Cram
MATH 204 tip sheet
1. Solution set of a system of equations
2. Matrix Operations
3. Evaluating Determinants
4. Vectors in 2 and 3 space
a) Dot product and it’s applications
b) Cross product and it’s applications
c) Calculating distances
d) Equation of planes
5. Vector spaces
a) Subspaces
b) Linear independence
c) Basis and Dimension
d) Row, column, and null spaces
e) Rank and nullilty
6. Eigenvalues, eigenvectors, and eigenspaces
7. Markov Chains
1. Solution set of a system of equations
1. Elimination method
A system of linear equations can be solved by either of Gaussian or GaussJordan
elimination methods
Gaussian elimination: The entire matrix should be reduced to reducedrowechelon
form by the following steps:
1. Find the first nonzero column. (starting from the left)
2. Exchange the top row with some other row, if necessary, to get a nonzero
value at the top of the column found above.
3. If the value that is now at the top of the column is not a one, divide the
whole row by it, to get a leading 1.
4. Add multiples of the top row to the rows below, to obtain only zeros below
the leading one.
5. Put the top row aside. GOTO back to the first step above until the entire
matrix is in row echelon form (the first 5 steps: Gaussian elimination
method).
6. Now from the last nonzero row and working upwards, add multiples of each
row to the rows above to obtain zeros on top the leading 1's (all 6 steps:
GaussJordan elimination method).
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2. Cramer's rule

If A is an
invertible matrix
( det (A)
≠
0) ,
then the solution to the
system
is given by,
where A
n
is the matrix found by replacing the n
th
column of A with
b
3. Writing an inverse of a matrix A as a product of elementary matricies E
Let A be an nxn invertible matrix. To write A
1
as a product of elementary matricies
E, do the following:
1. Apply the first step of row reduction to matrix A, then apply the same
iteration to matrix E, an identity matrix. The resulting elementary matrix
is called E
1
.
2. Apply the next iteration to A, and again apply those steps to another
elementary matrix E, another identity matrix. The resulting elementary
matrix is called E
2
.
3. Continue until A has been transformed to an identity matrix and at the
same time finding E
3
, E
4
, etc.
4. The product of the elementary matricies E
n
E
(n1)
…E
3
E
2
E
1
= A
1
For example, if A =
[1
2], then E
1
= [1
0], E
2
= [1
0], E
3
= [1
[3
4]
[3
1],
[0
.5]
Verify that E
3
E
2
E
1
= A
1
2. Matrix operations

If A and B are matrices whose sizes are such that the given operations are
.
2]
[0
1]