Question

Suppose the 2 × 2 matrix A has eigenvalues λ1 = 4 and λ2 = 3, with eigenvectors v1 and v2, respectively. If u = 5v1 + v2, then A 2u is equal to

(a) 25v1 + v2
(b) 25v1 + 3v2
(c) 80v1 + 9v2
(d) 100v1 + 3v2
(e) 400v1 + 9v2

Answer 1

`A^(2) u is equal to 80 v_(1)+9 v_(2)`

Answer: Option C

Explanation:

Given A is
`2 * 2` matrix has Eigen values
`\lambda_(1)=4` and \lambda_{2}=3 with Eigen vectors
`v_(1) \text { and } v_(2)` respectively.

`\lambda_(1)=4` and
`v_(1)` is the eigen vector, substitute this to A so then

`A v_(1)=\lambda_(1) v_(1)=4 v_(1)`

Squaring ‘A’ value ‘4’, we get

`A^(2) v_(1)=16 v_(1)`

Given
`u=5 v_(1)+v_(2)`, from this, the above can be written as

`A^(2)\left(5 v_(1)\right)=5 A^(2) v_(1)=5 * 16 v_(1)=80 v_(1)`

Similarly,
`\lambda_(2)=3` and
`v_(2)` is the eigen vector. Then,

`A v_(2)=\lambda_(2) v_(2)=3 * v_(2)`

Squaring ‘A’ value ‘3’, we get

`A^(2) v_(2)=9 * v_(2)`

To find
`A^(2) u`, multiply
`A^(2)` in both sides of the equation
`u=5 v_(1)+v_(2)`, we get

`A^(2) u=A^(2) *\left(5 v_(1)+v_(2)\right)`

`A^(2) u=5 A^(2) v_(1)+A^(2) v_(2)`

Substitute the value that found above,

`A^(2) u=80 v_(1)+9 v_(2)`