MATH-1201 Linear Algebra I

Final Exam Review

Your final exam covers material from the following sections:

Sections 1.1 - 1.10, Sections 2.1 - 2.5, Sections 3.1 - 3.5,and the handout on Complex Numbers that was given in class.

You are responsible for the proofs of the following theorems, butplease note that there may be proofs on your final exam and/orterm tests that are NOT on this list:

1. If A, B, and C are m × n matrices, then A + (B + C) = (A + B) + C

2. If A is m × n and B and C are n × p, then A(B + C) = AB + AC.

3. If A is an invertible n × n matrix, then its inverse is unique.

4. If A and B are invertible n × n matrices then

(

AT)

−1

=(

A−1)T

and (AB)−1 = B−1A−1

5. Let A be a square n × n matrix. Prove that A is invertible if and onlyif it can be written as a product of elementary matrices.

6. Let A be a square n × n invertible matrix. Prove the following:

(a) A adj(A) = (detA) In

(Note: To prove this you can use the Lemmathat was proved in class, but you don’t need to prove the Lemma.)

(b) A−1 =1

det(A)adj(A)

7. State and prove Cramer’s Rule.

8. Theorem 1, p. 63 of the 5th edition of Grossman’s text, namely:Let a, b, c be n-vectors and let α, β be scalars. Then

i. a · 0 = 0.ii. a · b = b · a.iii. a · (b + c) = a · b + a · c.iv. (αa) · b = α(a · b).

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9. Theorem 2, p. 242 of the 5th edition of Grossman’s text, namely:For two nonzero 2-vectors u and v, prove thatu · v = |u| |v| cos θ, where θ is the angle between the two vectors.

10. Theorem 2, p. 263 of the 5th edition of Grossman’s text, namely:Let u, v, w be vectors in IR3 and let α be a scalar. Then

i. u× 0 = 0 × u = 0.

ii. u× v = −(v × u).

iii. (αu) × v = α(u × v).

iv. u× (v + w) = (u× v) + (u × w).

v. (u× v) · w = (u · v) × w.

vi. u · (u × v) = v · (u × v) = 0.

vii. If u 6= 0 and v 6= 0, then u and v are parallel if and only ifu× v = 0.

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