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Apply the Gaussian elimination process to determine values of λ for which the
following linear system is consistent:
x − 3y + 4 = 0, 3x − 2y = λ, y = 6 − 2x .
Is Cramer’s Rule applicable for solving the linear system below? If yes, apply it.
Otherwise, alter the last equation in the system so that the solution can be obtained
by applying the Rule.
x + y + z = π
− πx + πy + √2z = 0
π^2x + π^2y+ 2z = 0
If a matrix has
n^2 entries, where n ∈ N , then it is a square matrix. Is it true or false? Justify your answer.
1a) what would be the gradient of a line parallel to the straight line 3x-y+4=0

b) what would be the gradient of a line perpendicular to the straight line

2. State whether the following pairs of lines whose equations are given are parallel, perpendicular or neither.

a. 2x-y+4=0 and 6x-3y+7=0
b. 7x+3y-8=0 and 3x-7y+1=0
c. x+3y-2=0 and 3x-y+4=0

3. Find the equation of the straight lines

a) Passing through the point (3, -2) and parallel to the line 4x-y+6=0

b) Passing through the origin and parallel to the line 5x+3y-7=0

c) Passing through the point (-2,5) and perpendicular to the line 3x-2y+8=0
1. Sketch the graph of the linear function

a) 2x + 3y = 6
b) 4y - 2x + 5 = 0

2. The following equations are in the general form Ax + Bx + C = 0. Express each of them in the form y = mx + b and state the gradient and y intercept of its graph

a) 3x - y + 2 = 0
b) 3x + 4y + 12 = 0
c) 2x + y - 3 = 0

3. The line PQ had an angle of inclination of 60 degrees. What is it gradient? ( answer in surd form)

4. What is the equation of the straight line having:
a) a gradient of 3 and a y intercept of -4
b) an angle of inclination of 135° and a y intercept of 5
x = u + t1v + t2w
What conditions on the vectors u, v, w ∈ R3, would create an object that is not a plane?
define T: R3→R3 by
. T(x, y, x)=(x+y, y, 2x-2y+2z)
Check that T satisfies the polynomial (x-1)square(x-2). Find the minimal polynomial of T.
Determine all values of the constant a for which the following system has (a) no
solution, (b) an infinite number of solutions, and (c) a unique solution.
ax1 + x2 + x3 = 1,
x1 + ax2 + x3 = 1,
x1 + x2 + ax3 = 1.
solve the system of equations
8x1 -x2 +2x3 =4
-3x1 +11x2 -x3 +3x4 =23
_x2 +10x3 -x4 =-13
_2x1 +x2 -x3 +8x4 =13
with x^(0) =[0 0 0 0]^T by using the Gauss Jacoboi and Gauss Seidel method. The exact solution of the system is x=[1 2 -1 1]^T. Perform the required number of iteration so that the same accuracy is obtained by birth the methods. What conclusions can you draw from the result obtained?
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