Transcript Chapter 4

Chapter 4
Matrices
By: Matt Raimondi
4-1 and 4-2 Introduction to
Matrices
• Matrices used when a group of numbers or variables are
blocked together.
• Each entry of a matrix is referred to as an element.
• The dimension of a matrix is defined by the number of rows by
the number of columns. So a 3 x 4 matrix would have 3 rows
and 4 columns.
• When a matrix is being multiplied by a scalar, any real number,
it is distributed to each element of the matrix.
• Matrices may only be added or subtracted if they have the
same dimensions. Then each element is added or subtracted
to the corresponding element of the other matrix.
4-1 and 4-2 Examples
• Here’s an example of scalar
multiplication.
• Each element was multiplied
by 4.
• Here’s an example of matrix
addition.
• Look how each element of A is
simply added to the
corresponding element of
matrix B.
• The only reason these two
matrices can be added is
because they have the same
dimension. Both are 3 x 2.
A
4A
B
A
A+B=
B
A+B
-4+2
-7 + (-5)
-9 + 7
-9+x
5+(-y)
-2+2
4-1 and 4-2 Introduction to
Matrices
• Two matrices are equal if and only if the following are true:
– They have the same dimensions.
– All of the elements of the first matrix are equal to the
corresponding elements of the second.
Lets look at some examples:

=

4-1 and 4-2 Solving Matrix
Equations
• The only way two matrices can be equal is if each element of
the corresponding matrix is equal. Let’s take a look at an
example problem:
5
X
3x
-6
5=5
X=7
=
5
7
21
-6
3x=21
-6=-6
To see if this is true we have
to solve for x.
If all of the statements
are true then the matrices are
equal.
Since the 2nd row 1st
column element says x=7 and
the 1st row 2nd column element is
3x=21, the matrices are equal.
4-1 and 4-2 Problems
• Solve the following matrix equations with the following
matrices A, B, and C:
1) -3A
2) B - A
3) 2B
4) B+A
5) A + 2B
6) 6B + 3A
7) Solve A=C for x and y.
Answers on next page.
4-1 and 4-2 Answers
1)
2)
3)
4)
5)
6)
7) x=8
y=5
4-3 Multiplying Matrices
• Unlike adding and subtracting matrices, they do not have to
have the same dimensions to be multiplied.
• The number of columns in the first matrix must match the
number of rows of the second matrix.
• The product will have the number of rows of the first matrix
and the number of columns of the second.
• For example, a 2x3 matrix multiplied by a 3x5 matrix will
produce a 2x5 matrix.
• If the number of columns of the first matrix does not match the
number of rows of the second, the product does not exist.
• To find the product, the rows of the first matrix are multiplied
by the columns of the second.
4-3 Examples
0(4)+3(0)
0(3)+3(3)
0(1) +3(2)
-1(4)+1(0)
-1(3)+1(3)
-1(1)+1(2)
• Lets take a look at the product
of these two matrices.
• The first is a 2x2 and the
second is a 2x3 matrix.
• Since the number of columns
of the first matrix matches the
number of rows of the second,
the product exists.
• Observe how each row of the
first matrix is being multiplied
by the columns of the second.
• The product will be a 2x3
matrix.
4-3 Problems
Given the dimensions of the matrices, state the size of the product if it
exists.
1) 4x8 * 8x2
4) 1x4 * 4x1
2) 3x3 * 3x2
5) 4x1 * 4x1
3) 3x2 * 3x3
6) 7x5 * 5x1
Compute the following products.
7)
Answers on next page
8)
9)
4-3 Answers
1) 4x2
2) 3x2
3) Non existent
4) 1x1
5) Non existent 6) 7x1
7)
8)
9)
4-4 Determinants
• All square matrices have a determinant. The only ones we will
deal with are 2x2 and 3x3 matrices.
• To find the determinant of a 2x2 matrix, you use the rule for
second order determinants.
•
ad-bc
• There are two ways to find the determinant of a 3x3 matrix.
• First is expansion by minors. To get your minors, pick any
row of the matrix look at the first digit, it will become a multiple
for a 2x2 matrix. Ignore the other numbers in the row and cover
up the other numbers in the column. You will only have 4
numbers left in the matrix and take that as a 2x2 matrix. Use
the digit you started with as a multiple of the matrix. Repeat
those steps for the remaining two digits in the row. Look at the
next page for the formula.
4-4 Determinants
• This is assuming you pick the first row for your multiples but
remember that you can pick any row.
= a*
-b*
+c*
• Observe how when a is the multiple, the corresponding matrix is
only the elements left over when the row and column of a are
covered.
• The other way to find the determinant of a 3x3 matrix is by using
diagonals. Look at the next page to see how it works.
4-4 Determinants
• First take the first two columns and add them on to the end.
• Then draw diagonals from the first entry of each row down and to
the right. You obtain aef, bfg, and cdh. Then start at the bottom
and draw diagonals up and to the right. You get gec, hfa, and idb.
• The determinant will equal aef+bfg+cdh-gec-hfa-idb.
4-4 Problems
Solve the following:
1)
2)
3)
Solve by expansion by minors:
4)
5)
6)
8)
9)
Solve by diagonals:
1) 14
Answers:
2)-22
8)-30
3)-22
9) 90
4)-32
5)-95
7)
6) 369
7)-323
4-5 Identity Matrix
• Similarly to multiplication of real numbers, matrices
have inverses and identities.
• We know that any number times 1 will give us the
original number. Likewise, any square matrix times the
identity matrix of equal size equals the original matrix.
*
=
4-5 Inverse of a Matrix
• In multiplication (1/a)*a = 1. This is the same as
(a-1)a = 1. The inverse of a number times itself
will equal 1. But with matrices, it will equal the
identity matrix.
A=
A* A-1= A-1*A= I2 =
A-1 =
4-6 Using Matrices to Solve
Systems of Equations
• A system of equations such as:
5x - 9y = -20
-4x + 3y = -5
can be solved using matrices.
• First, make a matrix of the coefficients.
• Then make a matrix of the variables and one for the
solutions.
*
=
4-6 Using Matrices to Solve
Systems of Equations
• To solve for x and y
we need to get rid of
the coefficient matrix.
• We can do this by
multiplying by the
inverse matrix on both
sides
A=
A-1 =