Chapter 1 - Systems of linear equations

Question 1

Definition : Elementary Matrices

Answer 1

An elementary matrix is always a square matrix. Any elementary matrix, which we often denote by E, is obtained from applying one row operation to the identity matrix of the same size. 4 Examples !!!

Question 2

Proposition 2.7 : Invertible

Answer 2

A matrix is invertible iff it is a product of elementary matrices.
pf:
=> Elementary matrices are invertible and the product of invertible matrices is invertible. Therefore, the product of elementary matrices is invertible.
<= If A is invertible, then there exists elementary matrices E_n…E_1 such that E_n …. E_1A = I <=> A = E_1^-1… E_n^-1 . Since the inverse of an elementary matrix is elementary QED #

Question 3

Proposition 4.3 : Properties of the det function

Answer 3

Let A ϵ M_n(Fbar). Then,
i) If B is obtained from A by multiplying the i-th column by a scalar λ then, det B = λ det A
ii) Suppose B & C differ from A only by the i-th column and the i-th column of A is the sum of the i-th columns of B and C. Then, det A = det B + det C.
iii) If B is obtained from A by exchanging 2 columns. Then, det B = -det A.
iv) det I = 1
v) det is the only function from M_n(Fbar) that has all of the above properties.
Note: Replace column with row and the prop still holds!!
PROOFS ON YELLOW SHEET (1.4.3)

Question 4

Proposition 4.4 : det A^t

Answer 4

det A^t = det A (A ϵ M_n(Fbar))

Question 5

Corollary 4.6 : det (AB)

Answer 5

det (AB) = detAdetB (A,B ϵ M_n(Fbar)). In particular, if A is invertible then detA is not equal to 0 and det(A^-1) = (detA)^-1

Question 6

Definition 4.7 : Minors, cofactors and adjugate

Answer 6

Yellow sheet 1.4.5