matrix powers-maths portfolio
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Matrix PowersIn mathematics, a matrix (plural matrices, or less commonly
matrixes) is a rectangular array of numbers, symbols, or expressions.
The individual items in a matrix are called its elements or entries. An
example of a matrix with six elements is-
Matrices of the same size can be added or subtracted element by element.
They are similar to vectors except that whilst columns of numbers
represent vectors, matrices are generally blocks of numbers made up of a
number of rows and columns. The rule for matrix multiplication is more
complicated, and two matrices can be multiplied only when the number of
columns in the first equals the number of rows in the second. In the
simplest form, matrices are often used to store information.
1)M= * + For n= 2,3,4,5,10,20,50
= * +
= * +
= * +
*
+ *
+
[ ( ) ( ) ( )( ) ( ) ( ) ]=
* +
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Mathematics -
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* +
=
* +
* +
=* + * +
=[ ( ) ( ) ( )( ) ( ) ( ) ]
=*
+
= * +
* + * +
* + * +
[ ( ) ( ) ( )( ) ( ) ( ) ]
=* +
= *
+
* + * +
* + * +
[ ( ) ( ) ( )( ) ( ) ( ) ]
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* +
= * +
= * + * +
=* + * +
[ ( ) ( ) ( )( ) ( ) ( ) ]
= * +
= * +
* + * +
=
= * + *
+
[ ( ) ( ) ( )( ) ( ) ( ) ]
= * +
= * +
Note:
*
+
*
+
*
+
*
+
NOTE: = Therefore
* + * +
* +
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= * +
=
* +
* +
* +
=* + *
+ *
+
= [ ( ) ( ) ( )( ) ( ) ( ) ] *
+
= * + *
+
= [ () ( ) ( )( ) ( ) () ]
= * +
There is a pattern that is observed. A geometric progression is seen as the value
ofn increases. The geometric progression is as follows- 2,4,8,16,32. This
can also be written as: , , , , Therefore the general term for is as follows-
= *
+Which can be written as:
= * +
Justification:
n=1
=
* +
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* +=
= 2
= * +
= * +
= * +n = 3
= *
+
= * +
= * +
2) P= * + S= * +
n = 3,4,5
= * += 2* +
=* +
= * +
* +
=* + * +
=[( ) ( ) ( ) ( )( ) ( ) ( ) ( )]
= *
+
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= * +
=*
+
= * + * +
= * + * +
=[( ) ( ) ( ) ( )( ) ( ) ( ) ( )]
= * +
= * +
=*
+
= * + * +
=* + * +
=[( ) ( ) ( ) ( )( ) ( ) ( ) ( )]
= * +
= * +
=* += 2 * +
n = 3,4,5
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= * +
=* + * +
=* + * +
=[( ) ( ) ( ) ( )( ) ( ) ( ) ( )]
= * += * +
= * +
= * + * +
= * + * +
=* +
= * +
= * +
= * + * +
= * + * +
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=* +
= *
+
The pattern observed:
Thus we get the following general term-
=* +
=* +
Let P/S = * +
Justification:
= * += R.H.S: 6+4=10
Therefore L.H.S=R.H.S
= * +
R.H.S: 28+ 8 = 36Therefore L.H.S=R.H.S
= * +
136 = 120 + R.H.S: 120+ 16
=136
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Therefore L.H.S=R.H.S
=* +
=R.H.S: 16+4=20
Therefore L.H.S=R.H.S
= * +
R.H.S: 104+ 8 =
112
Therefore L.H.S=R.H.S
= * +
656 = 640 + R.H.S: 640+ 16=656
Therefore L.H.S=R.H.S
3) A = * +
k=1A= * +
= * +k=2
A = * +
= * +
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k=3
A =
* +
=* +
k=4
A = * +
= * +
k=5
A = * +
=* +
k=6
A = * +
=
*
+
The pattern seen is:= * +
4) Thus substituting k=1,2,3 in the following general term-
= *
+
k=2 n=2
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Formula:= *
+
= [
]
= 2* +
=* +
k= -1 n=-1Formula:= * +
= [ ]
= -1* +
=
* +
k= 3 n=0Formula:= *
+
= [ ]
= * +
= * +
k=0 n=3Formula:= *
+
= [ ]
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There is no solution for the above matrix as there is no answer for zero raised
to any number.
Limitations of k and n
The general term hold true when the value of k and n are positive integers. The general term holds true when the value of k and n are negative integers The general term holds true when the value of n is zero and k is a positive or
negative integers.
The general term does not hold true when k is zero.Thus from the result we can conclude that the matrix is possible for all
values of n, positive integers, negative integers and zero.
The matrix is possible for positive and negative values of k. It is not possible
when k is equal to zero.