Question

For all positive integers n, determine an explicit expression for the matrix B^n, where B is a 2x2 matrix:

B = | 0 1 |
| -1 0 |

Answer 1

To find an explicit expression for the matrix B^n, compute B^2 and B^3 to observe a pattern. For even values of n, B^n = | (-1)^(n/2) 0 |, and for odd values of n, B^n = | 0 (-1)^((n-1)/2) |.

Step-by-step explanation:

Given the matrix B = | 0 1 |
|-1 0 |

We need to determine an explicit expression for the matrix B^n. To do this, we can compute B^2, B^3, and observe a pattern.

B^2 = B * B = | 0 1 | * | 0 1 | = |-1 0 |
|-1 0 |

B^3 = B * B^2 = | 0 1 | * |-1 0 | = | 0 -1 |
| 1 0 |

From the calculations, we can see that B^2 switches the signs and interchanges the entries in each row of the original matrix B.

By observing further, we can conclude that for even values of n, B^n = | (-1)^(n/2) 0 |
| 0 (-1)^(n/2) |

And for odd values of n, B^n = | 0 (-1)^((n-1)/2) |
| (-1)^((n-1)/2) 0 |