Question

Suppose a 2 × 2 matrix A has an eigenvector (1 2) , with corresponding eigenvalue −4. what is A(-2 -4) ?

Answer 1

`A ( (-2, -4) ) = ( (8,16) )`

Explanation:

If `lamda` is an eigenvalue with corresponding eigenvector `ul v` then:

`A ul v = lamda ul v`

From which we get with `lamda=-4` and `ul v = ( (1,2) )` :

`A ( (1,2) ) = -4 ( (1,2) ) \ \ \ ..... (star)`

A properties of matrices is that `Amu B = mu AB`, and so

`A ( (-2, -4) ) = (-2)A ( (1, 2) )`
`" " = (-2)(-4) ( (1,2) )` using `(star)`
`" " = 8 ( (1,2) )`
`" " = ( (8,16) )`

Answer 2

By definition, the eigenvectors (`mathbf e_i`) of `A`, and its eigenvalues (`lambda_i`) are related as:

`A mathbf e_i = lambda_i mathbf e_i`

Suppose a 2 × 2 matrix A has an eigenvector (1 2) , with corresponding eigenvalue −4. what is A(-2 -4) ?

This means that:

`A((1),( 2)) = - 4 ((1),( 2))`

So with the question what is A(-2 -4) ?:

`- 2 ( A((1),( 2)) )= - 2 ( - 4 ((1),( 2)))`

`implies A((-2),( -4)) = ((8),( 16))`