Linear Algebra Answers

Evaluate det(−A) and det(−A T ). Compare det(−A) and det(−A T ) for:

(2.1) A = -4 2

3 -3

(2.2) A = 3 1 -2

-5 3 -6

-1 0 -4

A=[3 0 2]

[4 -6 3]

[-2 1 8]

B=[-5 1 1]

[0 3 0]

[7 6 2]

C=[1 1 1]

[2 3 - 1]

[3 - 5 - 7]

Verify the following expressions(where possible and give reasons)

(i) A+(B+C) =(A+B) +C and A(BC) =(AB) C

(ii) (a-b) C=aC - bC and a(B - C) =aB - aC

(iii) (A^T) ^T=A and (A - B) ^T=A^T - B^T

Use Cramer's rule to solve the equation below:

y − z = 2

3x + 2y + z = 4

5x + 4y =1 

consider the following set of inequalitis: y_>5-2.5x
y_<3-x
x,y_>0
the correct graphical representation of this set of inequalities is given by

Consider the following augmented matrix 1 -1 2 1

3 -1 5 -2

-4 2 2x^2-8 x+2

Determine the values of x for which the system has

(i) no solution,

(ii) exactly one solution,

(iii) infinitely many solutions

Define: R^3→R^3 by

T(x,y,z)=(x-y+z,x+y,y+z)

Let v1= (1,1,1), v2= (0,1,1), v3= (0,0,1). Find a matrix of T with respect to the basis {v1,v2,v3}. Futher check T is invertible or not.

Evaluate det(−A) and det(−A^T). Compare det(−A) and det(−A^T) for:

(2.1) A = −4 2

3 −3 ;

(2.2) A = 3 1 −2

−5 3 −6

−1 0 −4

Assume that A is a 3 by 3 matrix such that det(A) = −10. Let B be a matrix obtained from A using the following elementary row operations:

R3 + 2R1 → R3,

5R1 → R1,

−2R2 → R2

R2 ↔ R3

Find the determinant of B obtained from the resulting operations, i.e., det(B).

Determine for which value (s) of k will the matrix below be non-singular.

(9.1) A = 2−k −3

2 k+1

(9.2) A = 2 2 1

3 1 3

1 3 k

Find A and write the following linear system in the matrix equation form (λI2 − A)X = 0

−x + y = λx

3x + y = λy

For the systems above, find:

(10.1) The determinant (known as the characteristic equation).

(10.2) Solve for λ when det(λI2 − A) = 0.

(10.3) Substitute for each value of λ from (ii) into the equation (λI2 − A)X = 0 and solve the corresponding system for X = x

y