Find a 3x3 matrix A such that Ax = 6x for all x in R^3.

Question

I'm entirely confused on how to approach this question. Any help would be appreciated!

Answer 1

Here's a hint: If the equation must be true for every x, then it must be true for the standard basis vectors in R^3. So take a generic 3 by 3 matrix A with entries a,b,c,d,e,f, etc. Next, replace x with the vector (1,0,0) and multiply out the equation. Repeat with x=(0,1,0), and then x=(0,0,1). This forces A to be 6*I, where I is the 3 by 3 identity matrix.

Let me know if you need more details. Good luck!