Skip to content
Class Notes

Module 4 Matrices

Overview of matrices: rectangular arrays of numbers, defining rows/columns and entries. Covers dimensions (m×n), locating entries, types (square, row, column matrices), and using matrices for computations (e.g., soccer equipment costs).

Preview 5 of 15 pages

See the first 5 pages exactly as they appear in the document.

  1. Page 1
    Read page 1 as text

    Finite M a t h M o d u l e 4: Matrices Reading: Matricesand Matrix Operations Two club soccer teams, the Wildcats and the M u d Cats, are hoping to obtain new equipment for an upcoming season. Table 1 shows the needs of both teams. Table 1 W i l d c a t s M u d Cats Goals 6 10 Balls 30 24 Jerseys 14 20 A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method i n which the data i n the soccer equipment table can b e displayed and used for calculating other information. Then, we will b e able to calculate the cost o f the equipment. Finding the Sum and Difference of Two Matrices To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A r o w in a matrix is a set of numbers that are aligned horizontally. A column i n a matrix is a set o f numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed i n [ ] or ( ) , and are usually named with capital letters. For example, three matrices named and are shown below. A=\matrix{{1}&{2}\\{3}&{4}} \displaystyleB = \begin{bmatrix}{1}&{2}&{7}\\{0}&{5}&{6}\\{7}&{8}&{2}\end{bmatrix} \displaystyleC = \begin{bmatrix}{1}&{3}\\{0}&{2}\\{3} & {1}\end{bmatrix} Describing Matrices A matrix is often referred to by its size or dimensions: m x n indicating m rows and n columns. Matrix entries are defined first by row and then by column. For example, to locate the entry i n matrix A identified as ay we look for the entry in rowj column In the matrix shown below, the entry in row 2, column 3 is a 2 3 = a_ll a_21 a_31 a_12 a_22 a_32 a_13" a_23 a_33

  2. Page 2
    Read page 2 as text

    A square matrix is a matrix with dimensions meaning that it has the same number o f rows as columns. The matrix above is an example of a square matrix. A row matrix is a matrix consisting o f one row with dimensions 1 x n a_ll a_21 a_31 . . . a_nl A column matrix is a matrix consisting of one column with dimensions m x 1 ct_ll a_21 a_31 A matrix may be used to represent a system o f equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further i n the next section, but first we will look at basic matrix operations. Matrices A matrix is a rectangular array of numbers that is usually named by a capital letter A, 8, C, and so on. Each entry i n a matrix is referred to as ay, such that represents the row and represents the column. Matrices are often referred to by their dimensions: m x n indicating m rows and n columns. Example 1 Given matrix find the dimensions of the given matrix and locating entries: 2 1 O’ A = 2 4 7 3 1 - 2

  3. Page 3
    Read page 3 as text

    What are the dimensions o f matrix A What are the entries at and o 3 1 and o 2 2 Solutions The dimensions are 3 x 3 because there are three rows and three columns. Entry a 3 1 is the number at row 3, column 1, which is 3. The entry a 2 2 is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column. Adding and Subtracting Matrices We use matrices to list data o r to represent systems. Because the entries are numbers, we can perform operations o n matrices. We add or subtract matrices by adding o r subtracting corresponding entries. In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add o r subtract a 3 x 3 matrix and another 3 x 3 matrix, but we cannot add o r subtract a 2 x 3 matrix and a 3 x 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix. A General Note Given matrices A and B of like dimensions, addition and subtraction of A and B will produce matrix C or matrix D o f the same dimension. A + B = C such that ay + by = CyA - B = D such that ay- by= dy Matrix addition is commutative. A + B = B + A It is also associative. Q4 + B) + C = A + (B + Q Example 2 Find the sum of A and B given , andB = “J u? Find the sum of

  4. Page 4
    Read page 4 as text

    A and B, given 4 3 1 2 5 0 9 7 andB = Find the difference of A and B Given A and B: ’ 2 - 1 0 -2" ' 6 10 -2" 4 = 14 12 10 andB = 0 - 1 2 - 4 4 - 2 2 - 5 2 - 2 Find the sum. Find the difference. Add matrix A and matrix B 2 6 ' ’ 3 - 2 A = 1 0 andB = 1 5 1 —3 - 4 3 Solutions N + |” e f] = [ a + <? b + f] d] hj [c 4- g d + hj Add corresponding entries. Add the entry in row 1, column 1, a 11( of matrix A to the entry i n row 1, column 1, of B. Continue the pattern until all entries have been added. \displaystyle{A}+{B}={\begin{bmatrix}{4}&{1}\\{3}&{2}\end{bmatrix}}+{\begin{bmatrix}{5}&{9}\\{0}&{7}\end{bmatrix}}={\begin{bmatrix}{4}+{5}& {1}+{9}\\{3}+{0}&{2}+{7}\end{bmatrix}}={\begin{bmatrix}{9}&{10}\\{3}&{9}\end{bmatrix}\right} We subtract the corresponding entries of each matrix.

  5. Page 5
    Read page 5 as text

    - 2 3 8 r — 2 - 8 3 - 1 ’ - 1 0 2 ’ 0 1 5 4 0 - 5 1 - 4 - 5 - 3 A— B = ' 9 - 1 0 - 2 ' ' 6 1 0 - 2 ' A + B = 1 4 1 2 1 0 + 0 - 1 2 - 4 4 - 2 2 - 5 2 - 2 Add the corresponding entries Subtract the corresponding entries ’ 2 - 1 0 - 2 ' ' 6 1 0 — 2" A — B = 1 4 1 2 1 0 — 0 - 1 2 - 4 4 - 2 2 — 5 2 - 2 2 6 ' ’ 3 - 2 ' 2 - 3 6 + ( - 2 ) ’ r 5 4" A + B = 1 0 + 1 5 — 1 - 1 0 + 5 = 2 5 1 - 3 - 4 3 - 3 + 3 —3 0_ Finding Scalar Multiples of a Matrix Besides adding and subtracting whole matrices, there are many situations i n which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication. Consider a real-world scenario i n which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed i n both labs. The school's current inventory is displayed i n Table 2. Table 2 Lab A Lab B Computers 15 27 Computer Tables 16 34 Chairs 16 34 Converting the data to a matrix, we have 15 27 34 34 CL2013 = 16 16 To calculate how much computer equipment will be needed, we multiply all entries in matrix C by 0.15, '(0.15)15 (0.15)27' (0.15)C_2013 = (0.15)16 (0.15)34 (0.15)16 (0.15)34 '2.25 2.4 2.4 4.05' 5.1 5.1

10 more pages in the full document

Unlock it once and it stays in your library, ready to chat with or turn into flashcards.

Unlock the full document

More Mathematics study materials

See all

Study this document with CramX

Once it is in your library, every tool can work from it.