## Problem 4

Let matrices $A, B \in M_{2}(\mathbb{R})$ such that $\operatorname{det} A=\operatorname{det}(A+B)=1$ and $\operatorname{det}(A-B)=3$, and the function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=\operatorname{det}(A+x B)$

a) Show that the function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=\operatorname{det} A+\alpha \cdot x+\operatorname{det} B \cdot x^{2}, \alpha \in \mathbb{R}$.

b) Find $\alpha$ and det $B$.

## Note

- Effective working time 3 hours.
- All problems are mandatory.
- 7 points are awarded for each correctly solved problem.


## National Mathematics Olympiad  Local stage -14.02.2015  11th grade $\mathbf{M}_{1}$  Solutions and marking schemes
## Problem 4

a) Verify by direct calculation.

b) $\left\{\begin{array}{l}f(1)=\operatorname{det}(A+B)=1 \Rightarrow 1+\alpha+\operatorname{det} B=1 \\ f(-1)=\operatorname{det}(A-B)=3 \Rightarrow 1-\alpha+\operatorname{det} B=3\end{array}\right.$

$\Rightarrow$ subtracting the relations we get $\alpha=-1$ and then det $B=1$

$.2 p$