Matrices and Systems of Linear Equations

Matrices and Systems of Linear Equations

The rows of A are its horizontal lines of entries, so A has three rows. The

columns of A are its vertical lines of entries, so A has two columns. The size

(also called order) is 3 × 2, its number of rows by its number of columns.

314. Recall that the element aij

is in the ith row and jth column of matrix A.

(A) a13 = 4

(B) a31 = –4

(C) a35 = –7

(D) a14 = 3

(E) a33 = 8

315. (A) A column vector is a matrix with only one column, Thus, the 3 × 1

column vector whose elements are 2, 3, and 8 is .

(B) A row vector is a matrix with only one row. Thus, the 1 × 4 row

vector whose elements are 0,–2, 1, 5 is [0 –2 1 5].

(C) For any square matrix A = [aij]n×n

, the main diagonal elements are

a11

, a22

, a33

, …, ann

Matrices and Systems of Linear Equations

. Note: Hereafter a matrix’s main diagonal will be

called simply its diagonal. Thus, the 3 × 3 matrix A = [aij]3×3 whose

diagonal elements are 3,–4, 5 and whose off-diagonal elements are

1’s is .

(D) I3×3 denotes the 3 × 3 identity matrix. An identity matrix is a square

matrix whose diagonal elements are 1’s and whose off-diagonal

elements are 0’s. Thus, I3×3 = .

(E) All the elements of the m × n zero matrix, denoted 0 (or 0m×n

), are

zero. Thus, the 2 × 3 zero matrix is .

316. Matrices can be added (subtracted) only if they are the same size. The sum

(difference) of two matrices is the matrix whose elements are the sums

(differences) of the corresponding elements of the two matrices.

317. Refer to question 316.

318. For a matrix A = [aij]m×n and a scalar k, kA = [kaij]m×n

is the product A by

k. Thus,

319. The product of a 1× n row vector and a n ×1 column vector is obtained by

multiplying corresponding elements and adding the resulting products. Thus,

Note: The product is undefined if the row and column vectors do not have the

same number of elements.

320. If A = [aij ]m×k

is a matrix of size m × k and B = [bij ]

k×n

is a matrix of size

k × n so that the number of columns of A is equal to the number of rows of B,

then the product AB is the m × n matrix C = [cij ]m×n whose ijth element is the

sum of the products of the corresponding elements of the ith row of A and the jth

column of B. When this inner matching of sizes occurs, the matrices A and B are

compatible for multiplication. Thus,

Note: The product AB is not defined if the number of rows of A is not equal to

the number of columns of B; and, in general, AB ≠ BA.

321. Proceed as in question 320.

322. Proceed as in question 321.

(A)

(B)

(C) No.

323. The determinant of a square matrix A, denoted det A or |A|, is a scalar

associated with the matrix and its elements. Note: The determinant is not

meaningful for non-square matrices.

A second order (2 × 2) determinant is defined as follows: det = ad –

bc.

Matrices and Systems of Linear Equations

Thus, = (–4)(2) – (5)(1) = –8 – 5 = –13.

Having a hard time figuring out how to do your assignment?

Ask our experts for help and get it done in no time!

The price is based on these factors:

Academic level

Number of pages

Urgency

Basic features

- Free title page and bibliography
- Unlimited revisions
- Plagiarism-free guarantee
- Money-back guarantee
- 24/7 support

On-demand options

- Writer’s samples
- Part-by-part delivery
- Overnight delivery
- Copies of used sources
- Expert Proofreading

Paper format

- 275 words per page
- 12 pt Arial/Times New Roman
- Double line spacing
- Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Delivering a high-quality product at a reasonable price is not enough anymore.

That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read moreEach paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read moreThanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read moreYour email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read moreBy sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more