[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.46,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.46,0:00:03.26,Default,,0000,0000,0000,,I want to make a quick\Ncorrection or clarification to
Dialogue: 0,0:00:03.26,0:00:06.54,Default,,0000,0000,0000,,the last video that you may or\Nmay not have found confusing.
Dialogue: 0,0:00:06.54,0:00:09.42,Default,,0000,0000,0000,,You may not have noticed it,\Nbut when I did the general
Dialogue: 0,0:00:09.42,0:00:13.09,Default,,0000,0000,0000,,case for multiplying a row by a\Nscalar, I had this situation
Dialogue: 0,0:00:13.09,0:00:17.24,Default,,0000,0000,0000,,where I had the matrix A and I\Ndefined it as-- it was n by n
Dialogue: 0,0:00:17.24,0:00:23.56,Default,,0000,0000,0000,,matrix, so it was a11, a12,\Nall the way to a1n.
Dialogue: 0,0:00:23.56,0:00:24.97,Default,,0000,0000,0000,,Then we went down this way.
Dialogue: 0,0:00:24.97,0:00:30.22,Default,,0000,0000,0000,,Then we picked a particular row\Ni, so we called that ai1,
Dialogue: 0,0:00:30.22,0:00:33.41,Default,,0000,0000,0000,,ai2, all the way to ain.
Dialogue: 0,0:00:33.41,0:00:36.06,Default,,0000,0000,0000,,And then we keep going down ,\Nassuming that this is the last
Dialogue: 0,0:00:36.06,0:00:40.25,Default,,0000,0000,0000,,row, so an1 all the\Nway to ann.
Dialogue: 0,0:00:40.25,0:00:42.50,Default,,0000,0000,0000,,When I wanted to find the\Ndeterminant of A, and this is
Dialogue: 0,0:00:42.50,0:00:46.77,Default,,0000,0000,0000,,where I made a-- I would call\Nit a notational error.
Dialogue: 0,0:00:46.77,0:00:51.36,Default,,0000,0000,0000,,When I wanted to find the\Ndeterminant of A, I wrote that
Dialogue: 0,0:00:51.36,0:00:55.46,Default,,0000,0000,0000,,it was equal to-- well, we could\Ngo down, and in that
Dialogue: 0,0:00:55.46,0:00:57.01,Default,,0000,0000,0000,,video, I went down this row.
Dialogue: 0,0:00:57.01,0:00:59.35,Default,,0000,0000,0000,,That's why I kind of highlighted\Nit to begin with,
Dialogue: 0,0:00:59.35,0:01:00.77,Default,,0000,0000,0000,,and I wrote it down.
Dialogue: 0,0:01:00.77,0:01:03.37,Default,,0000,0000,0000,,So it's equal to-- do the\Ncheckerboard pattern.
Dialogue: 0,0:01:03.37,0:01:06.61,Default,,0000,0000,0000,,I said negative 1\Nto the i plus j.
Dialogue: 0,0:01:06.61,0:01:07.64,Default,,0000,0000,0000,,Well, let's do the first term.
Dialogue: 0,0:01:07.64,0:01:16.24,Default,,0000,0000,0000,,I plus 1 times ai1 times\Nits submatrix.
Dialogue: 0,0:01:16.24,0:01:19.75,Default,,0000,0000,0000,,That's what I wrote in the last.\NSo if you have ai1, if
Dialogue: 0,0:01:19.75,0:01:22.81,Default,,0000,0000,0000,,you get rid of that row, that\Ncolumn, you have the submatrix
Dialogue: 0,0:01:22.81,0:01:24.55,Default,,0000,0000,0000,,right there: ai1.
Dialogue: 0,0:01:24.55,0:01:26.55,Default,,0000,0000,0000,,That's what I wrote\Nin the last video,
Dialogue: 0,0:01:26.55,0:01:27.97,Default,,0000,0000,0000,,but that was incorrect.
Dialogue: 0,0:01:27.97,0:01:31.05,Default,,0000,0000,0000,,And I think when I did the 2 by\N2 case and the 3 by 3 case,
Dialogue: 0,0:01:31.05,0:01:32.00,Default,,0000,0000,0000,,that's pretty clear.
Dialogue: 0,0:01:32.00,0:01:34.78,Default,,0000,0000,0000,,It's not times the matrix, it's\Ntimes the determinant of
Dialogue: 0,0:01:34.78,0:01:37.42,Default,,0000,0000,0000,,the submatrix, so this right\Nhere is incorrect.
Dialogue: 0,0:01:37.42,0:01:40.77,Default,,0000,0000,0000,,And, of course, you keep adding\Nthat to-- and I wrote
Dialogue: 0,0:01:40.77,0:01:44.52,Default,,0000,0000,0000,,ai2 times its submatrix\Nlike that.
Dialogue: 0,0:01:44.52,0:01:50.62,Default,,0000,0000,0000,,ai2 all the way to ain\Ntimes its submatrix.
Dialogue: 0,0:01:50.62,0:01:51.56,Default,,0000,0000,0000,,That's what I did\Nin the video.
Dialogue: 0,0:01:51.56,0:01:52.78,Default,,0000,0000,0000,,That's incorrect.
Dialogue: 0,0:01:52.78,0:01:56.25,Default,,0000,0000,0000,,Let me do the incorrect in a\Ndifferent color to show that
Dialogue: 0,0:01:56.25,0:01:57.68,Default,,0000,0000,0000,,this is all one thing.
Dialogue: 0,0:01:57.68,0:01:59.96,Default,,0000,0000,0000,,I should have said the\Ndeterminant of each of these.
Dialogue: 0,0:01:59.96,0:02:07.20,Default,,0000,0000,0000,,The determinant of A is equal\Nto minus 1 to the i plus 1
Dialogue: 0,0:02:07.20,0:02:16.18,Default,,0000,0000,0000,,times ai1 times the determinant\Nof ai1 plus ai2
Dialogue: 0,0:02:16.18,0:02:20.44,Default,,0000,0000,0000,,times the determinant of ai2,\Nthe determinant of the
Dialogue: 0,0:02:20.44,0:02:26.44,Default,,0000,0000,0000,,submatrix all the way to ain\Ntimes the determinant of the
Dialogue: 0,0:02:26.44,0:02:29.44,Default,,0000,0000,0000,,submatrix ain.
Dialogue: 0,0:02:29.44,0:02:31.89,Default,,0000,0000,0000,,It doesn't change the logic of\Nthe proof much, but I just
Dialogue: 0,0:02:31.89,0:02:33.91,Default,,0000,0000,0000,,want to be very careful that\Nwe're not multiplying the
Dialogue: 0,0:02:33.91,0:02:35.85,Default,,0000,0000,0000,,submatrices because\Nthat becomes a
Dialogue: 0,0:02:35.85,0:02:37.63,Default,,0000,0000,0000,,fairly complicated operation.
Dialogue: 0,0:02:37.63,0:02:38.26,Default,,0000,0000,0000,,Well, it's not that bad.
Dialogue: 0,0:02:38.26,0:02:38.82,Default,,0000,0000,0000,,It's a scalar.
Dialogue: 0,0:02:38.82,0:02:41.63,Default,,0000,0000,0000,,But when we find a determinant,\Nwe're multiplying
Dialogue: 0,0:02:41.63,0:02:43.36,Default,,0000,0000,0000,,times the determinant\Nof the submatrix.
Dialogue: 0,0:02:43.36,0:02:45.85,Default,,0000,0000,0000,,We saw that when we first\Ndefined it using the recursive
Dialogue: 0,0:02:45.85,0:02:48.76,Default,,0000,0000,0000,,definition for the n by n\Ndeterminant, but I just wanted
Dialogue: 0,0:02:48.76,0:02:51.23,Default,,0000,0000,0000,,to make that very clear.
Dialogue: 0,0:02:51.23,0:02:51.40,Default,,0000,0000,0000,,