Determinants

Let . For any , let be the matrix obtained from by deleting the th row and th column of . The -cofactor of , denoted by , is:

Let . Then the and cofactor of is

In , the determinant is the amount an area is scaled from a linear transformation. In , it is the change in volume after a linear transformation. For some , we refer to the "volume" as the "measure".

If the determinant is negative, then space was "flipped over".

A linear transformation with matrix is a dilation which scales by and by . The original area formed by the parallelogram was , and after the transformation it is . As such, the .

Here's something interesting. Recall that the magnitude of the cross product is also the area of a parallelogram formed by two vectors.

Although we have talked about parallelograms and rectangles, our work generalizes for any shape.

Let . The determinant of is:

You may sometimes see:

which is just shorthand for

Let . For any , we define the determinant of as:

which we refer to as the cofactor expansion of along the th row of .

Also, for any ,

which we refer to as the cofactor expansion of along the th column of .

Compute where .
Using cofactor expansion along the first row gives:

We could've chosen any row/column to do cofactor expansion along. Also, notice how for each number on the row, we are "crossing out" its column, then using the intersection of that column and row as our , and then using anything not crossed out as our other determinant.

Let
The cofactor matrix of is

Let . The determinant of is:

The adjugate is the transpose of the cofactor matrix

Let
The adjugate is:

Find if

Let . Then

(see identity matrix, Matrix Multiplication)

Let . is invertible if and only if , and in this case

Recall that the determinant of the area changed by a linear transformation. If this is 0, then the transformation squishes all vectors into a smaller space. Recall that a matrix is not invertible if it squishes all vectors into a smaller space. Otherwise, the inverse would not be a function.

Let and . Then

When you multiply every element of a matrix, you are basically "dilating" the matrix. Imagine if you had a 2x2 square, and you made it 4x4. The area does not change by merely a factor of 2. Instead, it changed by a factor of 4 (from 4 to 16), since . The idea here is the same.

Let . Then

(see Matrix Multiplication)

Since multiplication of real numbers is commutative, $$\det(AB) = \det(BA)$$

Since matrix multiplication is actually the composition of linear transformations, the change in area from the result of the two matrices is the same as the change in area from the first matrix, and then the change in area of the second matrix.

Let be invertible. Then

Recall that the determinant is the change in area or volume (generalized as the term "measure" for ). If the inverse of the matrix is "replaying" a linear transformation backwards, then it must also "reset" the area. This is done with the multiplicative inverse.

Let . Then

(see transpose)

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