This is archived information for Math 221 Sect 101 (Fall, 2002).
Here's a summary of what I remember talking about for the review session last night (Monday, November 4).
First, one student asked me to go over the answer to Quiz #6 problem 4. This was the (quite involved) proof of the Matrix Inverse Formula. Part 4(c) is quite difficult. Even the solution written up in the answer key is hard to follow, and that was the simplest way I could find to explain it. Parts 4(a) and 4(b) aren't so hard though, so it might be worth looking over the proofs of those. Part 4(b) especially is a nice theoretical application of the Invertible Matrix Theorem.
Then, there was a question about how row scaling and determinants interact. There are two ways of thinking about how a row scaling operation affects a determinant. They're equivalent, but it's maybe not easy to see why.
First, if matrix A is transformed into matrix B by a row scaling operation, say R1->3R1, then the determinant is transformed by multiplying by the same scalar. That is, the determinant of the result, B, is three times the determinant of the original matrix A: det(B)=3det(A). This is covered by Theorem 3.3 on page 187.
Second, as explained in the text further down page 187, you can "factor out a common multiple of one row" from a matrix in calculating determinants. That is, if one row of matrix C has a common factor, say 3, you can factor out the 3 from that row to produce a new matrix D, and det(C)=3det(D).
Even though it might seem these are saying the opposite thing, they are equivalent, as an example illustrates. If we have a matrix A=
| 1 | 2 | |||
| 3 | 4 |
| 1 | 2 | |||
| 9 | 12 |
However, we could view this instead as factoring a 3 out of a single row:
det
| 1 | 2 | |||
| 9 | 12 |
| 1 | 2 | |||
| 3(3) | 3(4) |
| 1 | 2 | |||
| 3 | 4 |
Unfortunately, this can become really confusing when you're comparing the above (scaling a row) with multiplying a whole matrix by a scalar multiple (scaling every row). Remember that, if A is transformed to B by the row operation R2->3R2, then det(B) = 3det(A)---that's what happens when one row is scaled by 3. Therefore, if you want to calculate det(3A), that's the determinant of the matrix 3A where every element is three times the corresponding element of A. Since this is the same as scaling every one of A's three rows, the determinant gets multiplied by 3 not once but three times! That is, det(3A) = 3(3)(3)det(A) = 27det(A).
For an n x n matrix A and any scalar r, the general formula is:
det(rA) = rn det(A)
That is, the determinant of the original must be multiplied by r to the power of the number of rows in A.
There was also a question about onto and one-to-one transformations. Remember that the usual way of determining whether a linear transformation is onto or one-to-one is to find its standard matrix and reduce that to echelon form to find its pivot positions. The transformation is onto if and only if there's a pivot position in every row. The transformation is one-to-one if and only if there's a pivot position in every column (or, equivalently, if and only if the matrix augmented with 0 has no free variables if and only if the columns of A are linearly independent).
Note that the work we've done with the Invertible Matrix Theorem and Quiz #5 problem #3 tells us everything we need to know about onto and one-to-one transformations. If we have a linear transformation with standard m x n matrix A then:
If A has more rows than columns (m>n), then it can't have a pivot position in every row though it can have a pivot position in every column. In other words, the transformation can't be onto, but it might be one-to-one.
If A has more columns than rows (n>m), then it can't have a pivot position in every column though it can have a pivot position in every row. In other words, the transformation can't be one-to-one, but it might be onto.
Finally, if A has the same number of rows and columns (that is, if A is a square, n x n matrix), then it either has fewer than n pivot positions, in which case it is neither one-to-one nor onto or it has exactly n pivot positions, so it is both one-to-one and onto.
Finally, I decided to do another diagonalization example, since we hadn't done too many of those. The example is available in a PDF file.
This is archived information for Math 221 Sect 101 (Fall, 2002).
Last revised: Wed Jan 29 12:48:08 PST 2003