Matrices Portfolio

Matrix powers

A matrix is a rectangular array of numbers (or letters) arranged in rows and columns. These numbers (or letters) are known as entries. Entries can be added and multiplied, but also squared. The aim of this portfolio is to investigate squaring matrices.

When we square the matrix M =  what we receive is a)  =  = .

Calculating the matrices for n = 3, 4, 5, 10, 20 and 50:

=

=

=

=

=

=

Examples shown above clearly indicate that while zero entries remain the same, non-zero entries change. Each entry is raised to a given power separately. Raising them to any power does not change the zero-entries.

Hence, we can observe that the general expression for the matrix  =  =

When considering the matrices P =  and S = , more conclusions can be drawn.

Raising each of them to the second power gives:

=  =  = 2

=  =  = 2

These matrices are simplified to 2 and 2 to make it easier to notice any patterns. Calculating Pn and Sn for other values of n we obtain:

=  =  = 2 = 4

=  =  = 2 = 4

=  =  = 2 = 16

=  =  = 2 = 16

=  =  = 512

=  =  = 512

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