Chapter 2 - Matrices I

Question 1

Explain what an m x n matrix is.

Answer 1

A rectangular array of m rows and n columns of numbers, known as entries.

Question 2

Two matrices are equal if and only if [_____].

Answer 2

Two matrices are equal if and only if they are the same size and their corresponding entries are equal.

Question 3

If A and B are matrices of the same size, the sum A + B is calculated by [_____].

Answer 3

If A and B are matrices of the same size, the sum A + B is calculated by adding corresponding entries.

Question 4

If A is any matrix and k is any number, the scalar multiple kA is the matrix obtained from A by [_____].

Answer 4

If A is any matrix and k is any number, the scalar multiple kA is the matrix obtained from A by multiplying each entry of A by k.

Question 5

If A, B and C are the same size,

A + B = [____] (Commutativity)
A + (B + C) = [____] (Associativity)

Answer 5

If A, B and C are the same size,

A + B = B + A (Commutativity)
A + (B + C) = (A + B) + C (Associativity)

Question 6

Define the transpose of a matrix.

Answer 6

The transpose of a matrix is the matrix formed by swapping the rows and columns of the original matrix.

Question 7

For any matrix A,

(A^T)^T = [\_\_\_]
(kA)^T = [\_\_\_]
(A+B)^T = [\_\_\_]

If A is an m x n matrix, A^T is an [____] matrix

Answer 7

For any matrix A,

(A^T)^T = A
(kA)^T = kA^T
(A+B)^T = A^T + B^T

If A is an m x n matrix, A^T is an n x m matrix.

Question 8

What does it mean for a matrix to be symmetric?

Answer 8

A^T = A if and only if the matrix is symmetrical.

Question 9

What is an ordered n-tuple?

Answer 9

An ordered sequence (a1, a2, a3, …, an) of real numbers.

Question 10

What is meant by ℝ^n?

Answer 10

A set of all the ordered n-tuples from ℝ.

Question 11

Given a system of linear solutions, we can consider the left hand side as [_____].

Answer 11

Given a system of linear solutions, we can consider the left hand side as a coefficient matrix A and the column x of variables, multiplied together: Ax.

Question 12

Let A = [a1, a2, a3, …, an] be an n x m matrix, written in terms of its columns a1, a2, …, an.

If x = vector{x1, x2, …., xn} is any n-vector, the product Ax is defined to be the m-vector given by:

[___________________]

Answer 12

Let A = [a1, a2, a3, …, an] be an n x m matrix, written in terms of its columns a1, a2, …, an.

If x = vector{x1, x2, …., xn} is any n-vector, the product Ax is defined to be the m-vector given by:

Ax = x1a1 + x2a2 + x3a3 + … + xnan

Question 13

Let A and B be m × n matrices, and let x and y be n-vectors in ℝ^n. Then,

A(x+y) = [\_\_\_\_\_]
A(ax) = [\_\_\_\_\_]
(A+B)x = [\_\_\_\_\_]

Answer 13

Let A and B be m × n matrices, and let x and y be n-vectors in ℝ^n. Then,

A(x+y) = Ax + Ay
A(ax) = (aA)x
(A+B)x = Ax + Bx

Question 14

Let x1 be any particular solution to the system to Ax = b.
Then, every solution x2 to Ax = b has the form:
[_____] for some solution x0 of the associated homogeneous system Ax = 0.

Answer 14

Let x1 be any particular solution to the system to Ax = b.
Then, every solution x2 to Ax = b has the form:
x2 = x1 + x0 for some solution x0 of the associated homogeneous system Ax = 0.

Question 15

Let Ax = b be a system of equations with augmented matrix [A | b]. Let rank A = r.

1. rank [A | b] is [____]
2. The system is consistent if and only if rank [A | b] = [____]
3. The system is inconsistent if and only if rank [A | b] = [____]

Answer 15

Let Ax = b be a system of equations with augmented matrix [A | b]. Let rank A = r.

1. rank [A | b] is r or r + 1
2. The system is consistent if and only if rank [A | b] = r
3. The system is inconsistent if and only if rank [A | b] = r + 1

Question 16

Define the dot product of the ordered n-tuples, (a1, a2, … , an) and (b1, b2, … , bn).

Answer 16

a1b1 + a2b2 + … + anbn

Question 17

Let A be an m × n matrix and let x be an n-vector. Then, each entry of the vector Ax is [_________].

Answer 17

Let A be an m × n matrix and let x be an n-vector. Then, each entry of the vector Ax is the dot product of the corresponding row of A with x.

Question 18

Define the identity matrix, I.

Answer 18

The n x n matrix with 1s in the main diagonal, and zeros elsewhere.

Question 19

Explain what is meant by the matrix transformation induced by A.

Answer 19

The transformation of a vector x, from ℝ^m to ℝ^n.

Question 20

Let A and B be matrices of sizes m x n and n x k respectively. Then, the (i, j)-entry of AB is [_____].

Answer 20

Let A and B be matrices of sizes m x n and n x k respectively. Then, the (i, j)-entry of AB is the dot product of row i of A with column j of B.

Question 21

Is matrix multiplication commutative?

Answer 21

No.

Question 22

If a is any scalar, and AB and C are suitably sized matrices,

IA = AI = [\_\_\_\_], where I is the identity matrix.
A(BC) = [\_\_\_\_]
A(B+C) = [\_\_\_\_] 
(B+C)A = [\_\_\_\_]
a(AB) = [\_\_\_\_]
(AB)^T = [\_\_\_\_]

Answer 22

If a is any scalar, and AB and C are suitably sized matrices,

IA = AI = A, where I is the identity matrix.
A(BC) = (AB)C
A(B+C) = AB + AC
(B+C)A = BA + CA
a(AB) = (aA)B = A(aB)
(AB)^T = B^T x A^T

Question 23

Summarise the process of block partitioning a matrix in order to simplify matrix multiplication.

Answer 23

The matrix is divided up into ‘blocks’, that can be substituted for letters representing matrices. The matrix can be multiplied and resubstituted. The blocks must be compatible for multiplication.

Question 24

Explain what a directed graph is.

Answer 24

A directed graph is a set of points (vertices) connected by arrows (edges). If the graph has n vertices [v1, v2, …, vn], then the adjacency matrix A = [a(ij)] is the n x n matrix whose (i,j)-entry is 1 if there is an edge from vj to vi and zero otherwise.

Question 25

If A is the adjacency matrix of a directed graph with n vertices, then the (i, j)-entry of A^r is [_____]

Answer 25

If A is the adjacency matrix of a directed graph with n vertices, then the (i, j)-entry of A^r is the number of r-paths vj to vi.