Mumbai University > Mechanical Engineering > Sem 4 > Applied Mathematics IV

Marks : 6 M

Year : May 2014

Let λ be the eigen value of matrix A

∴ $ \left| \begin{array}{ccc} -1-\lambda & 4 \\ 2 & 1-\lambda \end{array}\right| $ = 0

∴ (-1-λ) (1- λ)-8=0

∴ - $(1^2- λ^2)$-8=0

∴ $λ^2$-9=0

∴ λ= ±3

∴ Eigen values are 3 ,-3

Since matrix is of order 2X2 let $tan⁡A=a A+bI$-----(1)

Where a and b are constants.

We assume equation (1) is satisfied by eigen values λ

tan⁡λ=aλ+b-------------(2)

When λ=3

tan⁡3=a3+b-------------(3)

When λ=-3

-tan⁡(3)=-3a+b---------(4)

∴ Adding (3) and (4) we get ,

0 = 0 +b

∴ b = 0

From (3) tan (3) = 3a

∴ a = tan A / 3

Substituting a and b in (1)

tan A = $\dfrac{tan 3}{3}$ . A

∴ 3 tan A = A tan 3