Review exercise 1.23. Evaluate the following matrix expressions for and

and then find the general expression for the first two matrices for any .

Answer: We have

and

So the general equation appears to be

We can prove this by induction: Assume that the above equation holds true for some . Then for we have

So if the equation holds true for it holds true for as well. Also, if we define for any matrix then for we have

so by induction the equation above holds true for all .

Turning to the second matrix, we have

and

The entry of each matrix appears to be in general, but the entry is more complicated. The expression looks as if it might work; for this is and for this is .

We use induction to try to prove this. Assume that for some we have

We then have

So if the equation holds true for it holds true for as well. Also, for we have

so the equation above holds true for all .

Finally for the third matrix we have

Note that

so that the equation above holds true for also.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.