1.3 Matrix Operations

Question 1

What is a matrix?

Answer 1

A matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix.

Question 2

How do we describe the size of a matrix?

Answer 2

Always rows first, then columns.

rows x columns.

Question 3

What is a matrix with only one row called?

Answer 3

A row vector (or row matrix).

Question 4

What is a matrix with only one column called?

Answer 4

A column vector (or column matrix).

Question 5

A matrix with n rows and n columns we call…

Answer 5

A square matrix of order n.

Question 6

In a square matrix, A, the entries a11, a22, … , ann are collectively known as…

Answer 6

The main diagonal of A.

Question 7

What is the definition of two matrices being equal?

Answer 7

Two matrices are equal if they are of the same size and their corresponding elements are equal.
(A)ij = (B)ij for all values of i and j.

Question 8

How do we define matrix addition?

Answer 8

The sum A + B is the matrix obtained by adding the entries of B to the corresponding entries of A.
So, if A and B are the same size, and A = [aij] and B = [bij], then…
(A + B)ij = (A)ij + (B)ij = aij + bij
for all values of i and j

Question 9

How do we define matrix subtraction?

Answer 9

The difference A - B is the matrix obtained by subtracting the entries of B from the corresponding entries of A.
So, if A and B are the same size, and A = [aij] and B = [bij], then…
(A - B)ij = (A)ij - (B)ij = aij - bij
for all values of i and j

Question 10

How do we define scalar multiplication?

Answer 10

If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of A by c. The resulting matrix said to be a scalar multiple of A.
So, if A = [aij], then (cA)ij = c(A)ij = caij

Question 11

It is common practice to denote the scalar multiple (-1)B by…

Answer 11

-B

Question 12

What is the result of adding (or subtracting) two matrices of different sizes?

Answer 12

This operation is undefined.

Question 13

What is the size of the product of A and B, where A is m x r and B is r x n?

Answer 13

Size of AB is m x n.

Question 14

What is the requirement on the operands for matrix multiplication to be defined?

Answer 14

For AB to be defined, the number of columns on A must equal the number of rows on B.

Question 15

What is the size of the product of A and B?

Answer 15

AB, if defined, will have the number of rows of A x the number of columns of B.

Question 16

What is an augmented matrix?

Answer 16

It’s an abbreviation of a system of linear equations.

Question 17

The entry that occurs in row i and column j of a matrix A is usually denoted by…

Answer 17

aij

Question 18

How do we define matrix multiplication?

Answer 18

Given matrices A (size m x r) and B (size r x n), we find the entry ij of AB by taking the i-th row of A and the j-th column of B, multiply the corresponding entries of the row and column together, then sum the resulting products.

Question 19

How do we compactly denote the generalised contents of a matrix A?

Answer 19

[aij]

So, we can say A = [aij] and (A)ij = aij

Question 20

How do we define the transpose of a matrix?

Answer 20

The transpose results by interchanging the rows and columns. Formally:
(At)ji = (A)ij

Question 21

What do we observe about the transpose of a square matrix?

Answer 21

It can be obtained by ‘reflecting’ the entries about the main diagonal.

Question 22

How do we define the trace of a matrix?

Answer 22

Denoted by tr(A), is the sum of the entries along the main diagonal. Undefined if the matrix is not square.
Formally:
if A = [aij]nxn
tr(A) = a11 + a22 + a33 + … + ann

Question 23

What is tr(A)?

Answer 23

The trace of A - the sum of the entries along the main diagonal.

Question 24

What do we call the sum of the entries along the main diagonal of a square matrix?

Answer 24

The trace.