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Answer:
Step-by-step explanation:
Exponential growth function:
The exponential growth function for a population, after t years, is given by:
In which P(0) is the initial population and r is the growth rate.
A warren of rabbits has a population of 350. The population is increasing at a rate of 3% per year.
This means that
So
How can you write an exponential growth function to find the monthly growth rate?
Monthly growth rate means that the time is divided by 12(as one year has 12 months), so the equation will be:
Answer:
Step-by-step explanation:
REcall the following definition of induced operation.
Let * be a binary operation over a set S and H a subset of S. If for every a,b elements in H it happens that a*b is also in H, then the binary operation that is obtained by restricting * to H is called the induced operation.
So, according to this definition, we must show that given two matrices of the specific subset, the product is also in the subset.
For this problem, recall this property of the determinant. Given A,B matrices in Mn(R) then det(AB) = det(A)*det(B).
Case SL2(R):
Let A,B matrices in SL2(R). Then, det(A) and det(B) is different from zero. So
.
So AB is also in SL2(R).
Case GL2(R):
Let A,B matrices in GL2(R). Then, det(A)= det(B)=1 is different from zero. So
.
So AB is also in GL2(R).
With these, we have proved that the matrix multiplication over SL2(R) and GL2(R) is an induced operation from the matrix multiplication over M2(R).