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Linear Algebra
Use Gaussian elimination method to solve the following equation.
2x-3y+2z=6
x+4y-3z=12
3x-5y+2z=8
2x-3y+2z=6
x+4y-3z=12
3x-5y+2z=8
Linear Algebra
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 .
following linear system is consistent:
x − 3y + 4 = 0, 3x − 2y = λ, y = 6 − 2x .
Linear Algebra
Give examples, with justification, of the following:
ii) two non-singular 2× 2 matrices C and D , with |C| =√2 |D|
ii) two non-singular 2× 2 matrices C and D , with |C| =√2 |D|
Linear Algebra
Give examples, with justification, of the following:
i) two non-zero, 3×3 matrices A and B , with |A|=0, |B|=(5/7)i
i) two non-zero, 3×3 matrices A and B , with |A|=0, |B|=(5/7)i
Linear Algebra
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
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
Linear Algebra
If a matrix has
n^2 entries, where n ∈ N , then it is a square matrix. Is it true or false? Justify your answer.
n^2 entries, where n ∈ N , then it is a square matrix. Is it true or false? Justify your answer.
Linear Algebra
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
5x-2y-1=0
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
b) what would be the gradient of a line perpendicular to the straight line
5x-2y-1=0
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
Linear Algebra
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
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
Linear Algebra
x = u + t1v + t2w
What conditions on the vectors u, v, w ∈ R3, would create an object that is not a plane?
What conditions on the vectors u, v, w ∈ R3, would create an object that is not a plane?
Linear Algebra
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.
. 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.
