Introduction to Determinants - 3.1 and 3.2

In section 2.2, we defined det A for a 2x2 matrix. How can we generalize this for a square matrix with n > 2? What do determinants tell us?

The determinant of an nxn matrix A can be computed by…

A cofactor across any row or down any column

If A is a triangular matrix, then det A …

det A is the product of the entries on the main diagonal of A

Let A be a square matrix. How does row operations affect determinants?

Performing row operation on A to produce B:

• If a multiple of one row of A is added to another row to produce B, then

  det B = det A

• If two rows of A are interchanged to produce B, then

  det B = - det A

• If one row of A is multiplied by k to produce B, then

  det B = k det A

If a matrix has a row or column of all zeros, then…

det A = 0

A square matrix A is invertible if and only if…

det A does NOT equal zero

This is because A is invertible if A has n pivot positions (section 2.3), so det A cannot equal zero

If A is a nxn matrix, then det A^T …

det A^T = det A

The cofactor expansion of det A along the first row equals the cofactor expansion along the first column. That is, A and A^T have equal determinants.

If A and B are nxn matrices, then det AB …

det AB = (det A)(det B)