# Matrix Binomials IA

Matrix Binomials

Type 1 Internal Assessment

Matrices are rectangular arrays of numbers that are arranged in rows and columns, however the regular rules of algebra do not apply.

Let X= and Y= and calculate X2, X3, X4 ; Y2, Y3, Y4.

X2= × =

X3= × ×  =

X4= × ×  ×  =

Because matrices do not follow the algebraic rules of exponents, one can not simply distribute the exponent for each matrix value. Instead the matrix must be multiplied by itself however many times the exponent says. So for example, for X2, the matrix X must by multiplied with itself two times.

The pattern that has emerged is that with every increasing power the matrix value increases with an exponential power of 2.

21 is equal to 2 showed by the matrix . 22 is equal to 4 showed by the matrix  and similarly 23 is equal to 8 represented in the matrix .

Y2= × =

Y3= ×× =

Y4= ××× =

The pattern is very similar to the one above except that all the negatives in the original matrix will also become negatives.

Based on the results above we can conclude that for each exponent value Xn the matrix value will result in Xn-1 and the same goes for Yn resulting in Yn-1. Combining these terms would result in (X+Y)n-1.

(X+Y)n-1:

(X+Y)3-1 =

(X+Y)2 =

(X+Y)2 =                 X2+Y2

(X+Y)2 =                  which is the same as Xn-1+Yn-1

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Now let A=aX and B=bY where a and b are constants or scalars.

A=-1X

A2= -1        --1  =

A3= -1                        -1 =

A4= -1     -1  =

When multiplied by -1, the matrix values become negative but remain the same.

A=X

A2= -½        - =

A3= -½                        -½ =

A4= -½     -½ =

When multiplied by -½, the matrix values become negative. Also the values are divided by two (or halved).

A=½X

A2= ½        -½  =

A3= ½                        ½ =

A4= ½     ½ =

When multiplied by ½, the matrix values stay positive. Also the values are divided by two (or halved).

A=1X

A2= 1        -1  =

A3= 1                        1 =

A4= 1     1  =

When multiplied by ...