Matrix & linear equations

Question

The Matrix  

satisfies the following polynomial , 

then the value of K is ?

Answer 1

In the following solution I represent the matrix in tabular form:

X=

1	4
2	3

-2X=

-2	-8
-4	-6

X²=

1×1+4×2=9	1×4+4×3=16
2×1+3×2=8	2×4+3×3=17

+11X²=

99	176
88	187

X³=

9×1+16×2=41	9×4+16×3=84
8×1+17×2=42	8×4+17×3=83

-7X³=

-287	-588
-294	-581

X⁴=

41×1+84×2=209	41×4+84×3=416
42×1+83×2=208	42×4+83×3=417

-4X⁴=

-836	-1664
-832	-1668

X⁵=

209×1+416×2=1041	209×4+416×3=2084
208×1+417×2=1042	208×4+417×3=2083

X⁵-4X⁴-7X³+11X²-2X=

1041-836-287+99-2=15	2084-1664-588+176-8=0
1042-832-294+88-4=0	2083-1668-581+187-6=15

So K=

-15	0
0	-15

or -15I.

Comments

Sir, I'm just thinking to do it by some theorem.

But it's really a good approach & less time consuming :)  :)

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I doubt if there is a Theorem to cover it, but note the pattern as we increase the power of X. We get 

a	2b
b	a+b

where b is either 1 less or 1 more than a depending on whether the power is even or odd. Also a+b is 1 more or 1 less than 2b. So we only have to work out a and know the power of X to find all three remaining elements of the matrix. For example, X² has a=9; so b=8 (because the power 2 is even), 2b=16 and a+b=17.