Overall I had a pretty decent week on KA. I watched nine videos and understood the gist of about 75% of what Sal went through. I don’t think I studied for five hours though, so I’m disappointed about that. Part of me feels very motivated to get through these videos quickly with the New Year coming up and wanting to finish this unit before then, but when it comes to actually sitting down and going through the videos, I feel much less motivated. 🫤 I think a big factor is that I’m still finding it hard to visualize how matrix multiplication and operations work in general. It seems very vague and elusive to me, and so it’s hard to concentrate on the videos not knowing what the hell is going on. I know I’m slowly starting to understand the notation and how the math works, but I feel like I have to work SO hard to move forward just a tiny bit. My guess is that when I do wrap my head around all of this, it will all seem completely obvious. But as of now, it feels like I’m trying to read hieroglyphs. 🔍 🧐 🤔
Video 1 – Determinants Along Other Rows/Columns
As far a studying math goes, I thought this was a pretty cool video. 😎 And by that I mean 1) I actually understood what Sal was saying, and 2) the video explains a technique to make solving the determinant of a matrix which contains 0’s in it simpler. Just like last week, I still don’t think I have the words to explain the pattern you use to solve the determinant. It has to do with multiplying the elements across specific rows or columns by an expression of elements within a sub-matrix in the matrix itself… (That’s literally the best I can do to explain it, and that clearly made no sense. 😑) BUT, the point of this video is that picking out and using rows or columns that have 0’s in it means you can zero-out (as in cancel out) parts of the equation which makes the calculation that much simpler. Hard to explain, but satisfying to do.
Video 2 – Rule of Sarrus of Determinants
Before I explain what this video actually talked about, I should mention that the first picture of my notes above is the formula to calculate the determinant of a 3×3 matrix. I’m sure I’ve shown pictures of this formula from my notes in multiple posts before now, but here it is again. Even though I can’t put into words what the pattern is to solve the determinant, hopefully you can make sense of it by looking at that picture from my notes.
The picture of my second note above shows the pattern that Sal was talking about of another way to solve the 3×3 determinant which is called the Rule of Sarrus, which is a pretty sick name. You basically rearrange the matrix by swinging the bottom left and top right elements to the opposite side of their rows and multiplying the elements across all three diagonals and summing their products, THEN you do the same thing by moving the top right and bottom left elements to the opposite side of their respective rows, multiplying the diagonal elements but going the other direction, and finally subtracting all three products from the overall value. (Once again, that made no sense, but I’m hoping my notes make it clear what I’m trying to explain.)
Video 3 – Determinant When Row Multiplied by Scalar
Here’s what I wrote in my notes about this video:
“All this video is saying is that if you multiply a row in a matrix by a constant, k, the determinant will be k(ad – bc) [if it’s a 2×2 matrix], or in other words k*det(Matrix). If you multiply the entire matrix by k (i.e. every element in the matrix), the determinant will be kn where n is the number of rows in the matrix.”
Video 4 – (Correction) Scalar Multiplication of Row
In this video Sal made a correction to the formula from the previous video, but I didn’t understand the formula anyways so it went over my head. I know that the formula is the linear expression for the determinant of a matrix, but I don’t understand why the notation is written this way it is. 😔
Video 5 – Determinant When Row is Added
From my notes:
“The point of this video is if two matrices, say X and Y, have the exact same elements apart from one row each, a third matrix, Z, where the rows from X and Y are summed will have det(Z) = det(X) + det(Y). In other words, if X and Y are the same with one row being different, AND Z is the same as X and Y but adds the rows that are different from X and Y together, then det(Z) = det(X) + det(Y).”
(Once again, that probably didn’t make sense.)
Video 6 – Duplicate Row Determinant
“Sal’s explanation for why det(A) = –det(A) is if you can swap two rows in A without the elements changing (in other words, the rows have identical elements and are duplicates of each other), in his words, ‘if you have two duplicate rows, you could perform a row operation where you replace r1 = r1 – r2 and you’ll get a row of 0’s [if r1 = r2], and if you get a row of 0’s, you’re never going to be able to get the Identity Matrix’ (which is what you need for a matrix to be invertible and have a determinant).”
Video 7 – Determinant After Row Operations
This was the most confusing video I watched all week. I’m not going to try and explain what’s going on or how the math works (because I can’t), but in my notes I wrote, “the point (apparently) is that if A = B except one row in B subtracts another scaled row, ex. one row equals r3 – 3r2, the det(B) will still be equal to det(A) where the row simply equals r3 and there’s no row operation of r3 – 3r2.”
I don’t understand it, but I guess this is a big deal because it means you can use EROs on matrices that use this rule and know that the determinant will be the same. I don’t really see the big deal of this right now, but Sal made it seem like it’s a pretty important thing.
Video 8 – Upper Triangular Determinant
This video compared to the previous one was super satisfying. If my note above didn’t make it clear, the point of this video is that if you have an (n x n) matrix where all the elements below the diagonal are 0 (a.k.a. a reef matrix), you can calculate the determinant by simply finding the product of all the elements along the diagonal. Boom. Simple.
Video 9 – Simpler 4×4 Determinant
This was the final video I watched this week and it was an example of what the previous video was talking about, but I got confused on it. I paused the video before watching how Sal worked through it and tried to solve the determinant on my own by turning it into rref. The EROs I chose to do, however, were not the same ones that Sal did and we ended up with slightly different matrices at the end. As far as I can tell, I followed the ERO rules to turn it into rref but I didn’t get the same values in the diagonal as Sal did. I’m assuming I just have screwed up the EROs because it doesn’t make sense to me that I can follow the rules properly and get to a different solution. I didn’t have time to watch the full video, so I’ll start next week on this vid and will hopefully understand where I screwed up…
So all in all, not a terrible week. I have 12 videos left in this unit, including the one I ended this post with. I think I might be able to get through them all by next week but it will be tight. The good news is I’m off work this entire upcoming week so I don’t have any excuses in terms of being too busy. It’d be great to start the New Year off on a new unit, so fingers crossed I can lock in and make it happen! 🤞🏼