Let a and b be such that $-2A^2 + 4A -3I_2 = aA + bI_2$. To find $a + b$.

Question

$A \in M_2$ with characteristic polynomial $p(x) = x^2 -3x - 5$. Let a and b be such that $-2A^2 + 4A -3I_2 = aA + bI_2$. To find $a + b$.

Answer 1

Yes, this is correct. A simpler way to do it would be to use the Cayley-Hamilton theorem, i.e. the statement that each matrix satisfies its characteristic polynomial. From here, you can simply say that $A^2=3A+5$ and substitute this in.

Answer 2

A matrix satisfies its characteristic polynomial. $$-2A^2+4A-3I=aA+bI$$ $$-2A^2+(4-a)A+(-3-b)I=0$$ $$-2\left(A^2+\frac{a-4}{2}A+\frac{3+b}{2}I\right)=0$$ $$A^2+\frac{a-4}{2}A+\frac{3+b}{2}I=0$$ From this we need: $$\frac{a-4}{2}=-3$$ $$a-4=-6$$ $$a=-2$$ and $$\frac{3+b}{2}=-5$$ $$3+b=-10$$ $$b=-13$$ So $a+b=-15$