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MATH-2B Test 1 (take-home portion) Name:_____________________
Directions: Show all work on separate sheets of paper.
Due date: Wednesday, October 26, at the beginning of the class.
1. [8 pts.] Let consider a linear system.
.
a. Give the elementary matrices that reduce the coefficient matrix 
to row echelon form 
with 1’s on the main diagonal.
b. Use the elementary matrices to write a matrix in the form 
, where 
is a lower triangular matrix.
c. Explain how we can write directly by using “bookkeeping” of the elementary row echelon operations used in part a.
d. Use Forward substitution to solve the triangular system 
for the column 
.
e. Use Backward substitution to solve for 
the triangular system 
, where the column 
from part d.
f. Find 
-decomposition of the coefficient matrix , where is a lower unitriangular matrix (i.e. with ones on the main diagonal), 
is a diagonal matrix, and is an upper unitriangular matrix (i.e. with ones on the main diagonal).
g. Prove a general statement: if invertible 
matrix has -decomposition then it is unique, i.e. if
with matrices described in f. , then
, 
, 
.
2. [7 pts.] Let D(n) is the determinant of the nxn matrix with 7 on the main diagonal, 3 on the “diagonal” below the main diagonal and 4 on the “diagonal” above the main diagonal.
a. Write the recurrent equation of the 2nd order for D(n), D(n-1) and D(n-2) for n>2.
b. Find the general solutions of the equation from part a.
c. Specify the arbitrary coefficients A and B in the general solution from part b. using the initial conditions D(1)=7 and D(2)=37.