MAT 2401 Linear Algebra 3.1 The Determinant of a Matrix

HW Written Homework

Preview How do I know a matrix is invertible? We will look at determinant that tells us the answer.

If D=ad-bc ≠ 0 the inverse of is given by Recall Therefore, if D≠0, D is called the _________ of A

If D=ad-bc = 0 the inverse of DNE. Fact If D=0, A is singular. To see this, for a ≠ 0, we can do the following:

The Task Given a square matrix A, we wish to associate with A a scalar det(A) that will tell us whether or not A is invertible

Fact (3.3) A square matrix A is invertible if and only if det(A)≠0

Interesting Comments Interesting comments from a text: The concept of determinant is subtle and not intuitive, and researchers had to accumulate a large body of experience before they were able to formulate a “correct” definition for this number.

n=2 1. Notations: 2. Mental picture for memorizing

n=3

Q1: What? Do I need to remember this? Q2: What if A is 4x4 or bigger? Q3: Is there a formula for 1x1 matrix?

Observations

We need: 1. a notion of “one size smaller” but related determinants. 2. a way to assign the correct signs to these smaller determinants. 3. a way to extend the computations to nxn matrices.

Minors and Cofactors A=[a ij ], a nxn Matrix. Let M ij be the determinant of the (n-1)x(n-1) matrix obtained from A by deleting the row and column containing a ij. M ij is called the minor of a ij. Example:

Minors and Cofactors A=[a ij ], a nxn Matrix. Let C ij =(-1) i+j M ij C ij is called the cofactor of a ij. Example:

n=3

Determinants Formally defined Inductively by using cofactors (minors) for all nxn matrices in a similar fashion. The process is sometimes referred as Cofactors Expansion.

Cofactors Expansion (across the first column) The determinant of a nxn matrix A=[a ij ] is a scalar defined by

Example 1

Remark The cofactor expansion can be done across any column or any row.

Sign Pattern

Cofactors Expansion

Special Matrices and Their Determinants (Square) Zero Matrix det(O)=? Identity Matrix det(I)=? We will come back to this later….

Upper Triangular Matrix

Lower Triangular Matrix

Diagonal Matrix

Q: T or F: A diagonal matrix is upper triangular?

Example 2

Determinant of a Triangular Matrix Let A=[a ij ], be a nxn Triangular Matrix, det(A)=

Special Matrices and Their Determinants (Square) Zero Matrix det(O)= Identity Matrix det(I)=