# Matrix Binomials

John Vu

000948-111

Matrix Binomials

A matrix is a rectangular array of numbers arranged in rows and columns. Numbers or letters inside the brackets in matrices are called entries. Matrices can be added, subtracted, multiplied, divided, and also raised to a power. A common matrix can look like this  where a, b, c, and d are the entries.

Given the matrices X =  and Y =   I calculated X2, X3, X4; Y2, Y3, Y4.

Using my GDC (graphic display calculator) I evaluated these matrices.

X2 =                                                 Y2 =

X3 =                                                 Y3 =

X4 =                                               Y4 =

After calculating the powers of X and Y, I observed my solutions. Looking at this I saw a trend that emerged as I increased the power of the matrix. The trend was increasing the power of the matrix by one, caused the product to double its previous solution. When X is to the second power, the entries of the solution are all 2’s; when X is to the third power the entries are all 4’s; when X is to the forth power the entries are all 8’s, and so on. As for Y, the pattern is similar but some entries are negative (-). Using this information I produced an expression to solve for X to a certain power, and it is Xn =

letting “n” represent the power. For Y the expression is the same for the numbers in position of a and d. The expression for numbers of b and c is also the same but it is negative. As a matrix the expression is Yn =                         . I also calculated integer powers of (X + Y)n.

Using my GDC I found

(X + Y)1 =                                         (X + Y)3 =

(X + Y)2 =                                 (X + Y)4 =

When I finished with these calculations, I saw another pattern that was developed. Seeing this pattern I formed an expression to solve for the matrix (X + Y)n. For the numbers of a and d the expression is 2n and for c and d it is 0. As a matrix the expression is (X + Y)n =  .

To make sure all my expressions worked correctly, I calculated each matrix to the fifth power which ...