1 Use Cramer’s rule to determine the solution of the system of equations:

4x1 + 3x2 + 2x3 = 8

−x1 + 2x3 = 12

3x1 + 2x2 + x3 = 3

2. Consider the following system of equations involving a parameter k:

(2 − k)x1 + kx2 = 4

kx1 + (3 − k)x2 = 3

(a) For what value(s) of the parameter k does the system have a unique solution?

(b) Solve the system of equations using Cramer’s rule.

For each of the statements (i)—(iii), state whether it is true or false (i) Every system of linear equations has at least one solution. (ii) A system of four linear equations with three variables always has infinitely many solutions. (iii) One can determine whether two straight lines in R 3 intersect by solving an appropriate system of linear equations. (iv) Given any matrix A, then its reduced row echelon form is not unique. (v) Every elementary row operation can be undone by an(other) elementary row operation. (vi) One of the elementary row operations is to delete a row (i) A system of 3 linear equations with 4 variables cannot have a unique solution. (ii) It is possible to obtain two different reduced row echelon matrices from a given matrix by using two different sequences of elementary row operations. (iii) Elementary row operations on an augmented matrix do not change the solution set of the associated system of linear equations. (i) The matrix ? ? 1 0 a 0 100 0 0 0 7 ? ? is invertible for any value of a. (ii) If a system of linear equations with a square coefficient matrix A has infinitely many solutions, then det(A) = 0. (iii) Elementary row operations do not change the determinant of matrices. (i) A 2 × 2-matrix can have three distinct eigenvalues. (ii) If 0 is an eigenvalue of a square matrix A, then A is not invertible. (iii) Every 3 × 3-matrix can be diagonalized using elementary row operations.

prove that where a,b,c,d,e are real numbers with (a) not equal to zero. if this linear equation ax+by=c has the same solution set as this one : ax+dy=e, then they are the same equation.

Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W ) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T ]AB is a diagonal matrix where every entry along the diagonal is either a 0 or a 1.

A young couple buying their first home borrow $65,000 for 30 years at 7.4%, compounded monthly, and make payments of $450.05. After 5 years, they are able to make a one-time payment of $2000 along with their 60th payment.

(a) Find the unpaid balance immediately after they pay the extra $2000 and their 60th payment. (Round your answer to the nearest cent.)
$  

(b) How many regular payments of $450.05 will amortize the unpaid balance from part (a)? (Round your answer to the nearest whole number.)
payments

(c) How much will the couple save over the life of the loan by paying the extra $2000? (Use your answer from part (b). Round your answer to the nearest cent.)
$

Let ? be an eigenvalue of a matrix A. Explain why dim(?) ? 1

Problem 3. How many arrangements of MATHEMATICS are there that have ALL of the following properties:

TH appear together in this order and E appears somewhere before C?

Find an equation In slope intercept form passing through the point (-1,4) and parallel to the line 2x+2y=7 Show the steps

Given a right triangle with one leg length of a, one leg length of x-3 and the hypo is x, how does a affect x? Shown on a graph.

I solved for a, it is equal to the square root of 6x-9.... I am having a difficult time interpreting how a affects x.. I am not sure how to graph

Explain the difference between finding the horizontal and vertical asymptotes.

When will there be a point of discontinuity?

What could the graph look like when there is no horizontal or vertical asymptotes.

What is the best trick to be able to draw all polar curves within 5 minutes? Please provide detailed clarification and some examples.

Use f(x) = ?2x, g(x) = square root of x and h(x) = |x| to find and simplify expressions for the following functions and state the domain of each using interval notation. a . (h ? g ? f)(x) b. (h ? f ? g)(x) (g ? f ? h)(x)