MAT 2401 Linear Algebra

MAT 2401Linear Algebra 3.1 The Determinant of a Matrix http://myhome.spu.edu/lauw

HW • WebAssign 3.1 • Written Homework

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

Recall If D=ad-bc ≠ 0 the inverse of is given by

Fact If D=ad-bc = 0 the inverse of DNE.

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

n=3

Observations

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

Minors and Cofactors A=[aij], a nxn Matrix. Let Cij =(-1)i+jMij Cij is called the cofactor of aij.

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=[aij] is a scalar defined by

Example 1

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

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

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

Example 2

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

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

MAT 2401 Linear Algebra