Chapter 2 – Key Concepts

Question 1

Let A, B, and C be matrices of the same size, and let r and s be scalars. Evaluate the following w/ matrix algebra rules:

a. A + B
b. (A + B) + C
c. A + 0
d. r(A + B)
e. (r + s)A
f. r(sA)

Answer 1

a. A + B = B + A
b. (A + B) + C = A + (B + C)
c. A + 0 = 0
d. r(A + B) = rA + rB
e. (r + s)A = rA + sA
f. r(sA) = (rs)A

Question 2

If A is m×n and B is n×p with B consisting of the columns [b₁ … b_p] then what is the product AB?

Answer 2

AB is the m×p matrix whose columns are given by [Ab₁ … Ab_p]. That is, A [b₁ … b_p] = [Ab₁ … Ab_p].

Question 3

Each column of AB is a linear combination of what?

Answer 3

Every column of AB is a linear combination of the columns of A with weights given by the corresponding column of B.

Question 4

What is the row-column rule for computing AB?

Answer 4

The element of AB in the ith column and jth row is given by:

(AB)_ij = a_i1b_1j + … + a_inb_nj

Question 5

What is the relationship between the ith row of AB and the ith row of A?

Answer 5

row_i(AB) = row_i(A) B

Question 6

Let A be m×n and B and C have sizes that allow for the given multiplication to be defined. Then evaluate the following:

a. A(BC)
b. A(B + C)
c. (B + C)A
d. r(AB) for r ∈ ℝ
e. I_mA and AI_n

Answer 6

a. A(BC) = (AB)C
b. A(B + C) = AB + AC
c. (B + C)A = BA + CA
d. r(AB) = (rA)B = A(rB)
e. I_mA = AI_n = A

Question 7

Evaluate the following in terms of matrix algebra:

a. AB vs BA
b. AB = AC vs. B = C
c. AB = 0 vs. A or B = 0

Answer 7

In general…

a. AB ≠ BA
b. AB = AC does not mean B = C
c. AB = 0 does not mean A = 0 or B = 0.

Question 8

Evaluate: A^k for k ∈ ℝ.

Answer 8

A^k = A … A (k times)

Question 9

Evaluate the following:

a. (A^T)^T
b. (A + B)^T
c. (rA)^T
d. (AB)^T

Answer 9

a. (A^T)^T = A
b. (A + B)^T = A^T + B^T
c. (rA)^T = r(A^T)
d. (AB)^T = B^TA^T

Question 10

The transpose of a product of matrices is what?

Answer 10

The product of the transposes in reverse order.

Question 11

What does it mean for A to have an inverse A^-1?

Answer 11

There exists some matrix such that:

AA^-1 = A^-1A = I

Question 12

Say A = [[a b], [c d] ]. If ad-bc ≠ 0, what is A^-1?

What if ad-bc = 0?

Answer 12

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

If ad-bc = 0, then A is not invertible.

Question 13

What is the determinant of a 2×2 matrix A?

Answer 13

If A = [[a b], [c d] ], then det(A) = ad-bc

Question 14

If A is an n×n invertible matrix, then what must be true about the equation Ax = b for each b ∈ ℝⁿ?

Answer 14

The equation Ax = b has the unique solution x = A^-1b for all b ∈ ℝⁿ.

Question 15

Evaluate the following:

a. (A^-1)^-1
b. (AB)^-1
c. (A^T)^-1

Answer 15

a. (A^-1)^-1 = A
b. (AB)^-1 = B^-1A^-1
c. (A^T)^-1 = (A^-1)^T

Question 16

What is the inverse of the product of n×n matrices?

Answer 16

The product of the inverses of each matrix in reverse order.

Question 17

If an elementary row operation is performed on an m×n matrix A, how can we write the resulting matrix as a product of two matrices?

Answer 17

The resulting matrix is EA, where E is the m×m matrix created by performing the same row operation on I_m.

Question 18

Are all elementary matrices E invertible? If E does have an inverse, what is it?

Answer 18

Yes, their inverse is an elementary matrix of the same time that transforms E back into I.

Question 19

If an n×n matrix A is invertible, what must be true about its relationship to I_n in terms of row equivalence? If A is row equivalent to I_n, then how can we get A^-1 through elementary row operations?

Answer 19

A is row equivalent to I_n. The sequence of row operations that reduces A to I_n also transforms I_n into A^-1.

Question 20

Given A, how can A^-1 be found or determined not to exist with row operations on an augmented matrix?

Answer 20

Row reduce [A I]. If A is row equivalent to I, then [A I] is row equivalent to [I A^-1]. Otherwise, A does not have an inverse.

Question 21

What is the invertible matrix theorem?

Answer 21

Say A is a square n×n matrix. The following are equivalent statements (all true or all false):

a. A is invertible.
b. A ~ I_n.
c. A has n pivot positions.
d. Ax = 0 has only the trivial solution.
e. The columns of A form an L.I. set.
f. x ↦ Ax is one-to-one.
g. Ax = b has at least 1 soln. ∀ b ∈ ℝⁿ.
h. Columns of A span ℝⁿ.
i. x ↦ Ax maps ℝⁿ onto ℝⁿ.
j. ∃ n×n C s.t. CA = I.
k. ∃ n×n D s.t. AD = I.
l. A^T is invertible.

Question 22

If A and B are square invertible, what must be true about the relationships between A, A^-1, B, and B^-1 if AB = I?

Answer 22

B = A^-1 and A = B^-1.

Question 23

Let T : ℝⁿ → ℝⁿ and the std matrix of T be A. If T is invertible, what must be true about A? What is the inverse of T equal to?

Answer 23

A must be invertible.

The inverse of T is S s.t. S(x) = A^-1x and T and S satisfy the following:

1. S(T(x)) = x ∀ x ∈ ℝⁿ.
2. T(S(x)) = x ∀ x ∈ ℝⁿ.

Question 24

If A is m×n and B is n×p, what is the column-row expansion of AB?

Answer 24

AB = [col₁(A) … col_n(A)] [row₁(B) … row_n(B)] = col₁(A)row₁(B) + … + col_n(A)row_n(B)

Question 25

What does it mean for H to be a subspace of ℝⁿ?

Answer 25

H must have the following three properties:

a. 0 ∈ H.
b. ∀ u, v ∈ H, u + v ∈ H.
c. ∀ u ∈ H and c ∈ ℝ, cu ∈ H.

Question 26

What is the column space of a matrix A?

Answer 26

The column space of a matrix A is the set Col A of all linear combinations of the columns of A.

Question 27

What is the null space of a matrix A?

Answer 27

The null space of a matrix A is the set Nul A of all solutions of the homogeneous equation Ax = 0.

Question 28

The null space of an m×n matrix A is a subspace of what? Equivalently, the solution set of Ax = 0 is a subspace of what?

Answer 28

ℝⁿ

Question 29

What is a basis for a subspace?

Answer 29

A basis for a subspace H of ℝⁿ is a linearly independent set in H that spans H.

Question 30

What components of A form a basis for the column space of A?

Answer 30

The pivot columns of A.

Question 31

Let B = {b₁, …, b_p} be a basis for a subspace H. For a given x ∈ H, what are the coordinates of x relative to the basis B? What is the B-coordinate vector of x?

Answer 31

The coordinates of x relative to the basis B are the weights c₁, …, c_p s.t. x = c₁b₁ + … + c_pb_p.

The B-coordinate vector of x is: [x]_B = [c₁ … c_p].

Question 32

What is the dimension of a subspace?

Answer 32

The dimension of a nonzero subspace H, denoted by dim H, is the number of vectors in any basis for H. The dimension of the zero subspace {0} is defined to be zero.

Question 33

What is the rank of a matrix?

Answer 33

The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.

Question 34

What does the rank theorem say?

Answer 34

If a matrix A has n columns, then rank A + dim Nul A = n

Question 35

What does the basis theorem say?

Answer 35

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 36

What does the continued invertible matrix theorem say?

Answer 36

Let A be n×n invertible, then the following are equivalent (all true or all false) along with the rest of the IMT:

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