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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
A park has a stable population of birds. Prior to this situation, the birds’ population
increased from an initial low level. When the population of birds was 1000, the
proportionate birth rate was 40% per year and the proportionate death rate was 5% per
year. When the population was 3,000, the proportionate birth rate was 30% and the
proportionate death rate was 10%. Consider the population model under the following
assumptions:
(i) There is no migration and no exploitation.
(ii) The proportionate birth rate is a decreasing linear function of the population.
(iii) The proportionate death rate is an increasing linear function of the population.
Show that
The population grows according to the logistic model.
Find the stable population size.
If the shooting of birds is allowed at the rate of 15% of the population per year, find the
new equilibrium population.
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 byConsider 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.
Let 0<θ<2π, θ ≠ π. Consider the linear transformation T: C^2→C^2 given by matrix
[ cosθ -sinθ] (w.r.t standard basis)
[ sinθ cosθ]. Find the vector v1, v2 such that T v1= (e^iθ)v1, T v2= (e^-iθ)v2. Is {v1,v2} a basis for C^2? Give reason for your answer
Evaluate det(−A) and det(−AT). Compare det(−A) and det(−AT) 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).
