Use theorem 4.2.1 to determine which of the following are subspaces of mnn. (a) the set of all diagonal n × n matrices. (b) the set of all n × n matrices a such that det(a) = 0. (c) the set of all n × n matrices a such that tr(a) = 0. (d) the set of all symmetric n × n matrices. (e) the set of all n × n matrices a such that at = −a. (f) the set of all n × n matrices a for which ax = 0 has only the trivial solution. (g) the set of all n × n matrices a such that ab = ba for some fixed n × n matrix b

In the coordinate plan (-6,9) b (3,9) c (3,3) def is shown in the coordinate plan below

Browning labs is testing a new growth inhibitor for a certain type of bacteria. the bacteria naturally grows exponentially each hour at a rate of 6.2%. the researchers know that the inhibitor will make the growth rate of the bacteria less than or equal to its natural growth rate. the sample currently contains 100 bacteria.the container holding the sample can hold only 300 bacteria, after which the sample will no longer grow. however, the researchers are increasing the size of the container at a constant rate allowing the container to hold 100 more bacteria each hour. they would like to determine the possible number of bacteria in the container over time.create a system of inequalities to model the situation above, and use it to determine how many of the solutions are viable.

Acylinder has volume 45π and radius 3. what is it’s height?