This video is part 5 in a 6 part series on matrix math. None of these videos are for
any particular program. This part is covering the two topics of transpose and determinant.
To transpose a matrix is to essentially flip it. For example if we want to transpose matrix
"A" then what we have to do is make every single row a column and every column a row.
If the first column is 3 9 that means the new first row is 3 9. Second column is 4 7
which becomes the second row and 2 3 becomes the final row. It has the ability to turn
a row vector into a column vector. This means "B" will become the following. It is very
simple but it is easy to forget what that "T" means. If all we knew was that "A" equals
this matrix and we are given a home work assignment to give A^T that means that we are asked for
the transpose of matrix "A". A^T equals the matrix we created a moment ago. The determinant
is a little bit different. The determinant of a matrix is a single numerical value that
represents the value of the whole matrix. It has many different uses. For example it
can tell you if a matrix is singular. If a matrix is singular i means that it cannot
be inverted. The determinant can also tell you if the formulas or equations corresponding
to a given matrix have a unique single value. To calculate the determinant you have to combine
all of the numbers into one value. To practice we are going to use a 2 by 2 matrix. Our 2
by 2 matrix "C" is given here. Having the "C" with the straight lines next to it means
to find the determinant of it rather than to find the absolute value. To find the determinant
of "C" we first have to multiply all of the numbers in this direction which is 3 times
7 and subtract the numbers going in the other direction which is minus 4 times 9. That equals
-17. The determinant of "C" is -17. It gets much more complicated however when you work
with a larger matrix. This matrix called matrix "D" is more general. We are not going to use
numbers. Since it is a 3 by 3 matrix rather than a 2 by 2 matrix we have to go in this
direction 3 times. To make this slightly easier what I like to do is add two more columns
to the side that just repeat the first two columns. What I do is I go in one direction
all the way first. I have (a e i) all multiplied together then added to it the next diagonal
(e f g) and add to that the third diagonal of (c d h). Now subtract (c e g), subtract
(a f h) and subtract (b d i). The number that this returns is your determinant. If the determinant
equals 0 that means that there is no unique solution and the matrix is a singular matrix.
Calculating a determinant by hand can get extremely complicated very quickly. For example
our first matrix was a 2 by 2 matrix. The number of things that we had to multiply was
2 factorial which is 2 times 1. When we hit a 3 by 3 matrix that is 3! which is 6 things
to calculate. If you have a 4 by 4 matrix then you have to calculate 24 different diagonals.
A 5 by 5 would be 120. Luckily in both excel and matlab you have the ability to calculate
the determinant by telling the program to do it for you. You do not have to worry about
calculating 120 diagonals. You can just tell the program to give you the determinant of
the function.