Matrix Quick Study Guide
Vocabulary:
Augmented matrix 
Columns 
Commensurate matrices 
Commutative Property 
Dimensions 
Elementary Row Operations 
Elements 
Identity matrix 
Incommensurate matrices 
Inverse 
Invertible matrix 
Main Diagonal 
Matrix Addition 
Matrix Multiplication 
Matrix Subtraction 
Noninvertible matrix 
Nonsingular matrix 
Pivot 
Product Matrix 
Rectangular matrix 
Rows 
Scalar Multiplication 
Singular matrix 
Square matrix 
Zero Matrix 


Tasks:
_{ } ┌
┐ 
_{ } ┌
┐ 
_{ } ┌
┐ 
_{ } ┌
┐ 
1.
Find each of the
following (or mark them as incommensurate):
M_{1}+M_{1} 
M_{1}+M_{2} 
M_{1}+M_{3} 
M_{1}+M_{4} 
M_{1}+2M_{4} 
M_{1}×M_{1} 
M_{1}×M_{2} 
M_{1}×M_{3} 
M_{2}×M_{3} 
2.
Write I_{4}
(the 4x4 identity matrix).
3.
Write a singular
matrix that does not contain any 0’s.
4.
Find two square matrices
A and B that together show that matrix multiplication is noncommutative by
showing that A×B ≠ B×A.
5.
Find two square
matrices, A and B, A≠B, where neither A or B is the identity matrix, yet
A×B = B×A.
6.
Find two square
matrices, A and B, where neither A or B contains any 0’s, yet A×B is the zero
matrix (all zeroes).
7.
Without actually
inverting either matrix, show that M_{1}
and M_{4} are inverses of each other.
8.
Prove each of the
following statements (your proof may assume the 2x2 case):
a) I = I^{1}.
b) if (A = B^{1}) then (B = A^{1})
c) if (A×B = I) then (B×A = I)
d) (A×B)×C = A×(B×C) [ this is tedious ]
9.
Manually invert
the matrix M_{1} (label each step as an elementary row operation).
10.
Write a
nonsingular 3x3 matrix and manually invert it.
11.
Prove that, in
general, each Elementary Row Operation is really a Matrix Multiplication.
12.
In class, we
learned how to invert a matrix by constructing an augmented matrix and
transforming the lefthandside into the identity matrix using elementary row
operations, at which point the righthandside is the inverse. Prove that this works in general.
13.
Solve the
following system of linear equations using matrix inversion on your calculator:
3x + 2y – 5z = 22
8x – 5y = 3
x + 3y – 7z = 8
1.
Using matrix
inversion on your calculator, find the equation of a polynomial that contains
the points (1,2), (0,1), (1,4), and (2,17).
[ Hint: In general, you can fit k points with a (k1)degree
polynomial. Here, we have 4 points, so
we use a polynomial of degree 3. That
is, a cubic polynomial, of the form:
ax^{3} + bx^{2} + cx + d.]
2.
Using matrix
inversion on your calculator, prove: 1^{5} + … + n^{5} =
(1/12)(2n^{6 }+ 6n^{5 }+ 5n^{4} – n^{2}). Then, use your calculator to find 1^{5}
+ … + 7^{5} and show that this formula works when n=7.