Here are two more difference matrices, both important. But are they invertible? Cyclic $C = ? ? 2 ? 1 0 ? 1 ? 1 2 ? 1 0 0 ? 1 2 ? 1 ? 1 0 ? 1 2 ? ?$ Free ends $F = ? ? 1 ? 1 00 ? 1 2 ? 1 0 0 ? 1 2 ? 1 00 ? 1 1 ? ?$ . One test is elimination-the fourth pivot fails. Another test is the determinant, we don't want that. The best way is much faster, and independent of matrix size: Produce $x ? = 0$ so that $C x = 0$ . Do the same for $F x = 0$ . Not invertible. Show how both equations $C x = b$ and $F x = b$ lead to $0 = b_{1} + b_{2} + ? + b_{n}$ . There is no solution for other $b$ .

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Here are two more difference matrices, both important. But are they invertible? Cyclic

C = ? ? 2 ? 1 0 ? 1 ? 1 2 ? 1 0 0 ? 1 2 ? 1 ? 1 0 ? 1 2 ? ?

Free ends

F = ? ? 1 ? 1 00 ? 1 2 ? 1 0 0 ? 1 2 ? 1 00 ? 1 1 ? ?

. One test is elimination-the fourth pivot fails. Another test is the determinant, we don't want that. The best way is much faster, and independent of matrix size: Produce

x ? = 0

so that

C x = 0

. Do the same for

F x = 0

. Not invertible. Show how both equations

C x = b

and

F x = b

lead to

0 = b_{1} + b_{2} + ? + b_{n}

. There is no solution for other

b

.

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