Class Notes (834,935)
Canada (508,830)
Mathematics (1,919)
MATH 136 (168)
Lecture

Lecture 29.pdf

3 Pages
92 Views
Unlock Document

Department
Mathematics
Course
MATH 136
Professor
Robert Sproule
Semester
Winter

Description
Wednesday, March 19 − Lecture 29 : Determinants of elementary matrices. Concepts: 1. Recognize the effects the 3 elementary row (column) operations have on the value of the determinant of a matrix. 2. Recognize that det(A) = det(A ). T 3. Recognize conditions on A that force det(A) = 0. 4. Find determinants by using the properties of determinants. 29.1 Theorem − Suppose A and B are both n × n matrices. Then: Proof : - The proof of “A → cRi→ B implies det B = cdet A” is easily obtained by computing det B by expanding B along row R . i - The proofs of “A → Pij B implies det B = −det A” and “A → cRi+ Rj B implies det B = det A” can both be done by induction. To illustrate how this is done we prove “A → cRi+ Rj→ B implies det B = det A”. Base case: Let A be the 2 × 2 matrix whose first row is (a, b) and second row is (c, * d). Let Let A be the 2 × 2 matrix whose first row is (kc + a, kb+ d) (obtained by the ERO, kR 2 + R )1and second row is (c, d). * The same result is obtained if A is obtained by applying kR + R to A. So 1he 2 statement holds true for n = 2. Induction hypothesis: Let n be a natural number greater than two. Suppose the statement “A → cRi+ Rj→ B implies det B = det A” holds true for all matrices A m × m where m ≤ n – 1. Claim: “A → cRi+ Rj B implies det B = det A” holds true if A is n × n. Let A = [a ] be an n × n matrix (n ≥ 3) and A be obtained by applying cR + R to A. ij * * i j We compute det A by expanding along row R of A . where t ≠ it t ≠j. That is, * For each j = 1 to n, let M be thtjmatrix such that M is obtained by tjplying cR + R to i j M .tjere cR + R iefersjto rows belonging to M , but indexed as tjws of A where i, j ≠ t. * Now, for each j, M is an ij −1) × (n −1) matrix. By the induction hypothesis, det(M ) = ij det(M ).ijo we have Conclusion: By the principle of mathematical induction, “A → cRi+ Rj→ B implies det B = det A” holds true for A n ×
More Less

Related notes for MATH 136

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit