Chapter 2 theorems

Question 1

Theorem 1

Answer 1

Let A,B, and C be matrices of the same size, and let r and s be scalars.
a) A+B = B+A
b) (A+B)+C = A + (B+C)
c) A + 0 = A
d) r(A+B) = rA + rB
e) (r+s)A = rA + sA
f) r(sA) = (rs)A

Question 2

Theorem 2

Answer 2

Let A be an mxn matrix and let B and C have sizes for which the indicated sums and products are defined:
a) A(BC)=(AB)C
b) A(B+C) = AB + AC
c) (B+C)A = BA + CA
d) r(AB) = (rA)B = A(rB)
ImA = A = AIn

Question 3

Theorem 3

Answer 3

Let A and B denote matrices whose sizes are appropriate for the following sums and products.
a) (A^T)^T
b) (A+B)^T = A^T + B^T
c) For any scalar r, (rA)^T = rA^T
d) (AB)^T = B^TA^T

Question 4

Theorem 4

Answer 4

Let A is a 2x2 matrix = [{a,b},{c,d}]. if ad-bc != 0, then A is invertible and

A^-1 = 1/(ad-bc)*[{d,-b},{-c,a}]

Question 5

Theorem 5

Answer 5

If A is an invertible nxn matrix, then for each b in Rn, the equation Ax=b has the unique solution x = A^-1b

Question 6

Theorem 6

Answer 6

If A is an invertible matrix, then A^-1 is invertible and (A^-1)^-1 = A

b) if A and B are nxn invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. That is, (AB)^-1 = B^-1A^-1

c) If A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of (A^T)^-1 = (A^-1)^T

Question 7

Theorem 7

Answer 7

An nxn matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A^-1

Question 8

Theorem 8

Answer 8

Let A be a square nxn matrix. Then the following statements are equivalent:

a) A us an invertible matrix
b) A is row equivalent to the nxn identity matrix.
c) A has n pivot positions
d) The equation Ax = 0 has only the trivial solution
e) The columns of A form a linearly independent set
f) The linear transformation x-> Ax is one-to-one
g) The equation Ax = has at least one solution for each b in Rn
h) The columns of A span Rn
i) The linear transfomration x->Ax maps Rn onto RN
j) There is an nxn matrix C such that CA = I
There is an nxn matrix D such that AD = I
l) A^T is an invertible matrix

Question 9

Theorem 9

Answer 9

S(T(x))= x for all x in Rn
T(S(x)) = x for all x in Rn
Let T: Rn -> Rn be a linear transformation and let A be the standard matrix for T. Then T is invertible if and only if A is an invertible matrix. In that case, the linear transformation S given by S(x)=A^-1x is the unique function satisfying equation(1) and (2)

Question 10

Theorem 11

Answer 10

The null space of an mxn matrix A is a subspace of Rn. Equivalently, the set of all solutions of a system Ax = 0 of m homogenous linear equations in n unknowns is a subspace of Rn

Question 11

Theorem 12

Answer 11

The pivot columns of a matrix A form a basis for the column space of A

Question 12

Theorem 13

Answer 12

if matrix A has n columns
dim Col A + dim Nul A = n
rank A + dim Nul A = n

Question 13

Theorem 14

Answer 13

Let H be a p-dimensional subspace of Rn. Any linearly independent set of exactly p elements in H is automatically a basis for H. Also, any set of p elements of H that spans H is automatically a basis for H.

Question 14

Theorem 8 pt 2

Answer 14

m) The columns of A form a basis of Rn
n) Col A = Rn
o) dim Col A = n
p) rank A = n
q) Null A = {0}
r) dim NUl A = 0

Question 15

Theorem 1(ch 3)

Answer 15

The determinant of an nxn matrix A can be computer by a cofactor across any row or down any column. The expansion across the ith row using the cofactors is

det A = ai1Ci1 + ai2Ci2 + …+a1nC1n

Question 16

Theorem 2(Ch3)

Answer 16

If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A.

Question 17

Theorem3(Ch3)

Answer 17

Let A be a square matrix:
a) If a multiple of one row of A is added to another row to produce matrix B, then det B = det A

b) If two rows of A are interchanged to produce B, then det B = - det A

c) If one row of A is multiplied by k to produce B, then det B = kdetA

Question 18

Theorem 4(ch3)

Answer 18

A square matrix A is invertible if and only if det A !=0

Question 19

Theorem5(ch3)

Answer 19

If A is an nxn matrix then det A^T = det A

Question 20

Theorem6(ch3)

Answer 20

If and B are nxn matrixes, then det AB = (det A)(det B)