Colin Wick                  3/12/09

Period 7                Pre Calculus

Matrix Powers: Type I

        Matrices are useful mathematical tools that help us to interpret, represent, and ultimately understand information.  By comparing matrices in their original form, , we can observe specific patterns that are helpful in interpreting information demonstrated using matrices.  But, in some specific cases, the determinant of said matrices must be calculated in order to see a specific correlation.  This investigation analyzes relationships between both matrices and determinates of differing powers.  It further presents a general rule that can be used to calculate a specific pattern for any power.

        When considering the matrix: , an expression for the matrix of Mn in terms of n can be found by simply formalizing an observed pattern of the matrix Mn where n=1,2,3,4,5,6,7,8,9,10,20,50.  After plugging the matrix M into a graphing calculator, M to the previously stated powers can be easily calculated using the “^” key.  After making these calculations, the results obtained can be observed below in Table 1.1.

Table 1.1

An obvious pattern can be seen above, in which as the power n increases, so do the numbers in the top left portion (a) and lower right portion (d) of the matrix, while the other two values stay constant at zero.  In fact, the numbers in the top left and lower right increase exponentially.  Therefore, an equation can be derived to calculate Mn in terms of n:

Mn  = .  This equation accurately represents the exponential increase in the upper left and lower right quadrants of the matrix and, and is congruent with the pattern witnessed above.

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        Again, when considering the matrix M, another interesting relationship between the exponent, n, and the determinant of M can be observed and expressed in a general formula found by formalizing the observed pattern of the determinant of matrix Mn where n=1,2,3,4,5,6,7,8,9,10,20,50.  In order to find the determinants of the matrices listed in the second column of Table 1.1, they must first be entered into a graphing calculator.  Next, in the “Math” tab, found in the “Matrix” menu, the “det ( )” button can be used in conjunction with the matrix.  The results of this process for the matrices seen in Table ...

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