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.