Download Algebra 3 – Chapter 10 – Matrices

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Algebra 3 – Chapter 10 – Matrices
I. Terminology
Linear Algebra is a branch in mathematics that deals with matrices in such a way that an algebra of
matrices is permitted. In this section we provide a survey of how this matrix algebra is developed.
An m  n matrix, for m and n positive integers, is an array of the form (see a
a12
11
figure at the right). Each element (or entries) aij of the matrix is a real
a21 a22
number. An m  n matrix (read “m by n”) has m horizontal rows and n
a31 a32
vertical columns. A matrix of m rows and n columns is called a matrix


with dimensions. Note that the row dimension is always listed first. A
row matrix is any matrix with dimensions 1 n . A column matrix is any
am1 am2
matrix with dimensions m 1 .
a13
... a1n
a23
... a2 n
a33
... a3n


am3 ... amn
The matrices used to represent systems of linear equation are called augmented matrices.
If an m  n has the same number of rows as columns, that is if m = n, the matrix is referred to as a square
matrix. Two m by n matrices A and B are said to be equal, written as A = B provided that each entry bij in
A is equal to the corresponding entry bij in B.
The Identity Matrix is a square matrix with 1’s down the diagonal and 0’s for every other entry.
1 0 0
1 0
I1  [1]
I3  0 1 0
I2 
0 1
0 0 1
To add/subtract matrices, we add/subtract the corresponding elements. The matrices must have the same
dimensions. Capital letters are used to denote matrices. A matrix having zeros for all of its members or
elements is called a zero matrix and is often denoted as O. When a zero matrix is added to another
matrix of the same dimensions, that same matrix is obtained. Thus a zero matrix is an additive identity
( A  0  A ).
Dimensions of a Matrix
The dimension of a matrix is the number of rows by the number of columns. Dimension of matrix A is
2  3 (two “by” three), because there are 2 rows and 3 columns. Rows always come first!
1 2 3
A
4 5 6
Augmented Matrix
Take the coefficient of the system of equations and place them in a matrix in the same corresponding
positions.
3x  4 y  7 z  0
3
4
7
0
 2 0 6 9
1 1 2 1
x  y  2 z  1
2 x  6 z  9
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II. Matrix Operations
Adding and Subtracting Matrices
In order to add or subtract matrices, you simply add or subtract each corresponding entry. However, the
dimensions of each matrix must be the same! So in the A+B we are going to take the first entry of matrix
A (3) and add it to the first entry in matrix B (-1). This results in a two (2). You continue this process for
each entry. The same is true for subtraction. For example, in order to find A  B , you take the entry in A
and subtract the corresponding entry in B.
3 2
0
1
4
Let A  0 5 .5 and B  5
1 1 
2
3
2 .5
0
A B 

3   1 2  4
03
2
2
3
05
5  2
.5.5 = 5
3
1
1 2
1  0   
3 1 2
Scalar Multiplication
The number being multiplied by the matrix is multiplied to each entry inside the matrix. In order to find
2 A  B , you need to multiply matrix A by 2. Then take the matrix 2A and subtract matrix B.
3 2
0
9 6
0
.
3A = 3 0 5 .5 = 0 15 15
3 3 3
1 1 
Multiplication of Matrices
Multiplication of matrices is not commutative which means the order does matter. The number of columns
in the first matrix must equal the number of rows in the second matrix. If A has dimensions m  n and B
has dimensions n  p (notice the columns of A equal the rows of B), then matrix AB (after you multiply)
will have dimensions m  p . [We will work examples in class.]
III. Systems of Equations
System of Equations is a collection of two or more equations. A solution of a system of equations
consists of values for the variables that are solutions to each equation of the system. We use substitution,
elimination, Row Echelon Form, Cramer’s Rule, and Inverses to solve systems of equations. It is possible
to have one, no, or infinitely many solutions.
Row Echelon Form – has 1’s down the diagonal and zeros below it. The letters represent other numbers.
Reduced Row Echelon Form – has 1’s down the diagonal and zeros as all other entries.
Row Echelon Form
1 a b d
0 1 c e
0 0 1 f
Reduced Row Echelon
1 0 0 a
0 1 0 b
0 0 1 c
We use Row Operations in order to get a matrix is Row Echelon Form.
1. Interchange any two rows.
2. Multiple or divide any row by a nonzero constant.
3. Add or subtract any two rows. (Then replace a row with the new equation.)
Complete the following assignment.
Pg. 771 # 1-6 (concepts), #4, 8, 12 (problems)
Pg. 802 Read Historical Feature (not the problems).
Pg. 803 #1-6 (problems)
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Solving a System of Linear Equations Using Matrices
A matrix is in row echelon form when
1. The entry in row 1, column 1 is a 1, and 0’s appear below it.
2. The first nonzero entry in each row after the first row is a 1; 0’s appear below it, and it appears to
the right of the first nonzero entry in any row above.
3. Any rows that contain all 0’s to the left of the vertical bar appear at the bottom.
Matrix Method for Solving a System of Linear Equations (Row Echelon Form)
1. Write the augmented matrix that represents the system.
2. Perform row operations that place the entry 1 in row 1, column 1.
3. Perform row operations that leave the entry 1 in row 1, column 1 unchanged, while causing 0’s to
appear below it in column 1.
4. Perform row operations that place the entry 1 in row 2, column 2, but leave the entries in columns
to the left unchanged. If it is impossible to place a 1 in row 2, column 2, then proceed to place a 1
in row 2 column 3. Once a 1 is in place, perform row operations to place 0’s below it. [If any
rows are obtained that contain only 0’s on the left side of the vertical bar, place such rows at the
bottom of the matrix.]
5. Now repeat step 4, placing 1 in the next row, but on column to the right. Continue until the bottom
row or the vertical bar is reached.
6. The matrix that results is the row echelon form of the augmented matrix. Analyze the system of
the equations corresponding to it to solve the original system.