Sorry, this question is making my head spin. Can someone help?
If x is an eigenvector for the square matrix A corresponding to eigenvalue 5, ie. if Ax = 5x, evaluate A^2x + 11Ax + 3x in terms of x.
Ax = 5x --->
A^2 x = (A A)x = A (A x) = A (5x) =
5 A x = 25 x
No problem! I can help you with that.
To evaluate A^2x + 11Ax + 3x in terms of x, we can use the given information that x is an eigenvector for matrix A corresponding to eigenvalue 5.
We know that Ax = 5x.
Let's start with A^2x:
A^2x = A(Ax) = A(5x) = 5(Ax)
Since Ax = 5x, we can substitute it back:
A^2x = 5(5x) = 25x
Next, let's evaluate 11Ax:
11Ax = 11 * 5x (substituting Ax = 5x)
= 55x
Finally, let's evaluate 3x:
3x
Now we can put it all together:
A^2x + 11Ax + 3x = 25x + 55x + 3x
= (25 + 55 + 3)x
= 83x
So, A^2x + 11Ax + 3x in terms of x is 83x.