$A=$ $\left[ {\begin{array}{cc} 0 & -1\ 1 & 0 \end{array} } \right] $

Subject: Applied Mathematics 4

Topic: Matrices

Difficulty: Medium

Characteristic equation of A is $\mid A-\lambda I \mid=0$

$\lambda^2-(-1+1)\lambda+\mid A \mid=0$

$\lambda^2-0+(-1-8)=0$

$\lambda^2-9=0$

$\lambda^2=9$

$\lambda=\pm3$

Let $\phi(A)=3tanA$

Consider, $\phi(A)=\alpha_1 A+ \alpha_0 I$

$3tanA=\alpha_1 A+ \alpha_0 I$ (1)

$3tan\lambda=\alpha_1 A+ \alpha_0 $ (2)

for $\lambda=3$ we get

$3tan3=\alpha_1 3 +\alpha_0$ (3)

and for $\lambda=-3$ we get from (2),

$3tan(-3_=\alpha_13+\alpha_0$

$-3tan3=-3\alpha_1+\alpha_0$ (4)

Adding (3) and (4) we get,

$0=0+2\alpha_0$

$\alpha=0$

From (3) we get $3tan3=\alpha_1 3$. So, $\alpha_1=tan3$

From (1), we get $3tanA=tan3A$. So, $Atan3$