SOLUTION: Ok - I am having problems with solving the follwoing equation -- can you please help me out. Please note that the M amounts within the brackets are in sets of four -- 2 on top of 2

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Question 28312: Ok - I am having problems with solving the follwoing equation -- can you please help me out. Please note that the M amounts within the brackets are in sets of four -- 2 on top of 2 - for example M= a 0(being on the top line while 0 b are right below it -- lining up the a and the 0 and then the 0 and the b.
Let M= [a 0]
[0 b] with a, b real numbers.
a) Solve that M(squared) =MM= [a(squared) 0]
[0 b(squared)].
b) What is det(M)?
c) Under what condition(s) does M(inverse) exist?
d) What is M(inverse)?
e) Suppose M= [2 0]
[0 1]. What M(tenth power)? (hint: find M(squared), M(cube).... is there apattern?
Answer by venugopalramana(3286) (Show Source):
You can put this solution on YOUR website!
Let M= [a 0]
.......[0 b] with a, b real numbers.
a) Solve that M(squared) =MM= [a(squared) 0]
..............................[0 b(squared)].
MM=*
=
=
b) What is det(M)?
=a*b-0*)=ab
c) Under what condition(s) does M(inverse) exist?
det(m)=ab should not equal zero...that is either a or b or both should not be zero
d) What is M(inverse)?
step1..replace each element with their minor
minor matrix=
step2....multiply each element with (-1)^(m+n),where m and n are row and column numbers.
cofactor matrix=
step3......transpose the above
transpose =
step 4...divide by determinant to get inverse
inverse of M = (1/ab)*
e) Suppose M= [2 0]
..............[0 1]. What M(tenth power)? (hint: find M(squared), M(cube).... is there apattern?
USING ABOVE WE GET
MM=M^2=
MMM=M^3=*
=
YES THERE IS A PATTERN
M=
M^2=
M^3=
...............................HENCE
M^10=