I a) If A and B are two square matrices of the same size, prove that \( \operatorname{det}\left[\begin{array}{ll}A & B \\ B & A\end{array}\right]=\operatorname{det}(A+B) \operatorname{det}(A-B) \) b) Let \( M=\left[\begin{array}{ll}P & Q \\ R & S\end{array}\right] \) where \( \mathrm{P}, \mathrm{Q}, \mathrm{R} \) and S are square matrices of the same size and P is invertible. Find a matrix N of the form \( \left[\begin{array}{ll}A & O \\ B & C\end{array}\right] \) such that \( M N=\left[\begin{array}{ll}I & P^{-1} Q \\ O & S-R P^{-1} Q\end{array}\right] \) Hence show that if P and R commute then \( \operatorname{det} M=\operatorname{det}(P S-R Q) \)

1 Petrol is stored in enormous cylindrical tanks at an oil refinery. These tanks can have volumes of up to 2000 kl . a How many litres of petrol can be stored in a 2000 kl tank? b A petrol tanker has a capacity of 1600 litres. How many loads of petrol can be filled from a full storage tank?

The polynomial of degree \( 5, P(x) \) has leading coefficient -5 , has roots of multiplicity 2 at \( x=1 \) and \( x=0 \), and a root at \( x=-5 \). Write a function for \( P(x) \) in factored form, and also write \( P(x) \) expanded in general form. a) Factored form: \( P(x)= \)

Sean los intervalos \( A=(-3,6], \quad B=(0,9) . \quad C=[7,12) \). Hallar: \( A \cup B, A \cap B, A \cup C, A \cap C, B \cup C, B \cap C \)

1. \( \frac{x}{4}-\frac{1}{3}=-\frac{4}{12}-x \)

The best experimental design (1) alters only one condition between the controls and the experimental condition (2) includes a large sample size and a control, and alters only one condition between the controls and the experimental condition (3) includes a control (4) includes a large sample size for each condition