Question

I need help with an IB Math Internal Assessment on Matrix Powers.

Answer 1

To clarify : Consider the Matrix M = (2 0 (on top) and 0 2 (on bottom)) Calculate M^n for n=2,3,4,5,10,20,50. Describe in words any pattern you observe. Use this pattern to find a general expression for the matrix M^n in terms of n.

Answer 2

There are other questions but I would really appreciate it if I could just have help with this one because I don't even know where to begin.

Answer 3

I suspect you know what an identity matrix is, In two dimensions
1 0
0 1

Now if you multiply by that identity matrix, you do not change the original matrix

1 0
0 1
times
1 0
0 1
is still
1 0
0 1
so I am going to make a leap of faith and say
2 0
0 2 to the nth is

2^n 0
0 2^n

Answer 4

Try the first few :)

Answer 5

proof is like this by the way
2 0
0 2 is the same as

2 times the identity matrix

so when we multiply we multiply the scalar twos and the matrix part stays the original

1 0
0 1

so we end up with
2^n times the identity matrix

Answer 6

There is a system you can download called "Matlab". I only have an ancient DOS (yes some of us still use it) version but I am sure if you search you can find it. If they have a cheap version it is perfect for fooling with matrix math.

Answer 7

Thank you so much! I didn't even think of using an indentity matrix. I also used the matlab - which was a great help! Thank you again!

Answer 8

n+12=4