Subject: Applied Mathematics 2

Topic: Matrices

Difficulty: Medium

Let $cosA =α_1A + α_0I$ ……………………………..(1)

Where $α_1$ & $α_0$ are the constants to be determined ,

to find $α_1$ & $α_0$

we suppose that eqn (1) must be satisfied by

λ i.e. $cos λ =α_1 λ + α_0$…………………..(2)

To find eigen values of A ,characteristic equation is , |A- λI| =0

Or $ \begin{vmatrix} π-λ&\frac{π}{4} \\ 0&\frac{π}{2}-λ \end{vmatrix} $ =0 , expanding we get (π-λ) ($\frac{π}{2}$-λ) =0

i.e. λ = π , $\frac{π}{2}$ putting both the values of λ in eqn (2) we get

$cos π =α_1 π + α_0$ or $-1=α_1 π + α_0 $……………..(3)

$cos \frac{π}{2} = α_1 \frac{π}{2} + α_0$ or $0= α_1 \frac{π}{2} + α_0 $……………(4)

Solving eqns (3) & (4) we get $α_0 = 1$ & $α_1=-\frac{2}{π}$

Putting these values in (1) ,we get

$cos A = -\frac{2}{π} \begin{bmatrix} π&\frac{π}{4} \\ 0&\frac{π}{2} \end{bmatrix} +1 \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} = \begin{bmatrix} -1&-\frac{1}{2} \\ 0&0 \end{bmatrix} $