Subject: Applied Mathematics 2

Topic: Matrices

Difficulty: Medium

Let $A^{50} = α_1A + α_0I$ ……………………………..(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. $λ^{50} = α_1 λ + α_0$…………………..(2)

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

Or $ \begin{vmatrix} 2-λ&3 \\ -3&-4-λ \end{vmatrix} $ =0 , expanding we get $λ^2 + 2λ+1 =0$

i.e. λ = -1 , -1 putting λ=-1 in eqn (2) we get, $1= - α_1 + α_0 $………………………………..(3)

Since eigen values are repeated to get next equation, differentiate eqn (2) & put λ=(-1)

i.e. $50λ^{49} = α_1 $, by putting λ=-1 we get, $α_1 = - 50$ ,

by putting $α_1 = - 50$ in (3) we get $1= 50 + α_0$ i.e. $α_0= - 49$ ,

by putting above values in (1) we get,

$ A^{50} = -50 \begin{bmatrix} 2&3 \\ -3&-4 \end{bmatrix} -49 \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} = \begin{bmatrix} -149&-150 \\ 150& 151 \end{bmatrix} $