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 A2x + 11Ax + 3x in terms of x.
Of course, I can help you with that! To evaluate the expression A2x + 11Ax + 3x in terms of x, we will need to understand some concepts related to eigenvectors and eigenvalues.
First, let's recall the definition of an eigenvector. An eigenvector of a square matrix A is a non-zero vector x such that when we multiply it by A, the result is a scalar multiple of x. In other words, Ax is a scalar multiple of x. This scalar is called the eigenvalue corresponding to the eigenvector x.
In this case, we are given that x is an eigenvector of A corresponding to eigenvalue 5, which means that Ax = 5x.
Now, let's evaluate the expression A2x + 11Ax + 3x using this information. To do that, we need to simplify each term separately.
1. A2x: To calculate A2x, we can start by noting that A2 = A * A. This means that A2x = A(Ax). Since we are given that Ax = 5x, we can substitute it to get A2x = A(5x). Using matrix multiplication, we obtain A2x = 5(Ax). Since Ax = 5x, we have A2x = 5(5x). Therefore, A2x simplifies to 25x.
2. 11Ax: Similar to the previous step, we can substitute Ax = 5x to get 11Ax = 11(5x). Simplifying this expression further, we have 11Ax = 55x.
3. 3x: This term doesn't involve any matrix multiplication, so it stays the same.
Finally, we can sum up all the simplified terms to get the overall expression in terms of x:
A2x + 11Ax + 3x = 25x + 55x + 3x = 83x.
Therefore, the expression A2x + 11Ax + 3x simplifies to 83x in terms of x.