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Let M 8 11 Find formulas for the entries of Mn, where n is a positive integer M-

User Igouy
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1 Answer

5 votes

Answer:

Mn[1,1] = 8^n + 11^n, Mn[1,2] = 8^n + 11^n, Mn[2,1] = 8^n + 11^n, Mn[2,2] = 8^n + 11^n

Explanation:

To find formulas for the entries of the matrix Mn, where n is a positive integer, we need to start with the given matrix M:

M = | 8 11 |

Now, let's calculate a few powers of M to see if we can identify a pattern:

M^2 = M * M

M^2 = | 8 11 | * | 8 11 |

M^2 = | (88 + 118) (811 + 1111) |

M^2 = | 64 + 88 88 + 121 |

M^2 = | 152 209 |

M^3 = M * M^2

M^3 = | 8 11 | * | 152 209 |

M^3 = | (8152 + 11152) (8209 + 11209) |

M^3 = | 1216 + 1672 1672 + 2299 |

M^3 = | 2888 3971 |

Let's observe the pattern in the calculations above. The entries of Mn seem to depend on the powers of 8 and 11. Specifically, the (1,1) entry is a sum of powers of 8, and the (1,2) entry is a sum of powers of 11. Similarly, the (2,1) entry is a sum of powers of 8, and the (2,2) entry is a sum of powers of 11.

Here are the formulas for the entries of Mn:

Mn[1,1] = 8^n + 11^n

Mn[1,2] = 8^n + 11^n

Mn[2,1] = 8^n + 11^n

Mn[2,2] = 8^n + 11^n

These formulas will give you the entries of the matrix Mn for any positive integer n.

User Jeff Argast
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