Given a matrix, state its dimensions.
Given two matrices determine whether you can add them, and perform the addition if possible.
Is matrix addition commutative?
Given two matrices determine whether you can multiply them, and perform the multiplication if possible.
Name the method used for matrix multiplication.
Is matrix multiplication commutative?
Write the identity matrix for a given square dimension.
Write the relationship between a matrix A, its inverse, and the identity matrix.
Given a 2x2 square matrix with entries a, b, c, d, find the determinant.
Given a 2x2 square matrix with entries a, b, c, d, find the inverse using the determinant and the rearranged matrix.
Given a system of two equations in two variables, wrtie it in matrix notation with a coefficient matrix, a variable matrix, and a constant matrix.
Given a coefficient matrix, a variable matrix, and a constant matrix, write a corresponding system of linear equations.
Explain how to solve a matrix equation A*X = B.
Multiply a matrix by the identity matrix and show your work.
Friday, October 28, 2011
Thursday, October 27, 2011
10-27 TI-83 graphing calculator matrix screens
When you first access the MATRIX menu the screen that appears has the names of matrices.
NAMES | MATH | EDIT |
 |  |  |
1. The NAMES menu is used to paste the name of a matrix into the home screen or a program.
2. The MATH menu contains all of the operations that can be done with a matrix.
3. The EDIT menu is where you set the size of the menu and enter the elements.
The example below will take you through the basic steps of entering a matrix and using it to solve a system of equations, which is one of the main applications of matrices that you will use in an algebra course.
Example: Use a matrix to solve the following system of equations.
To use a matrix to solve a system of equations, you put the matrix in reduced row echelon form. While this can be a tedious and time consuming process to do by hand, the TI-83 will do this in one step.
Go to MATRIX and EDIT. |  |
Define [A] to be a 3X4 matrix. Use the cursor and the ENTER key to move through the spaces. |  |
Now enter the coefficients and constants from the system by entering the value and pressing ENTER. |  |
Return to the home screen (   ) and accessthe MATRIX MATH menu and scroll down to item B. |  |
Hit ENTER to return to the home screen. |  |
Enter matrix [A] by pressing  1 and close the parentheses. |  |
Press ENTER and read the solution as x=2, y=-1, z=4. |  |
Matrices can be entered directly into the home screen by using the [ ] keys which are accessed by enteringand
. You enter [ to start the matrix and then enclose each row in [ ] and close the matrix with ]. Below are some screens showing some manipulations with matrices.
 |  |  |
10-27 Inverse matrices on a graphing calculator
2nd MATRX > > EDIT ENTER
(see MATRIX [A] blink x2)
2 ENTER (puts in number of rows)
> > (moves cursor to blink)
2 ENTER (puts in the number of columns)
> > (moves cursor to highlight entry at 1,1)
1 ENTER (puts the value 1 in row i column 1)
(cursor moves to highlight entry at 1,2)
1 ENTER (puts the value 1 in row 1 column 2)
(cursor moves to highlight entry at 2,1)
0 . 4 ENTER (puts the value 0.4 in row 2 column 1)
(cursor moves to highlight entry at 2,2)
0 . 1 5 ENTER (puts the value 0.15 in row 2 column 2)
2nd MATRX (see highlight NAMES and highlight 1: )
ENTER (see [A] on new screen)
X-1 (see [A]-1)
ENTER (see [[-.6 4 ] [1.6 -4]] )
Continue item 12c on page 63 by entering the values for matrix B [[ 3000][ 840]]
2nd MATRX (see highlight NAMES and highlight 1: )
ENTER (see [A] on new screen)
X-1 (see [A]-1)
ENTER (see [[-.6 4 ] [1.6 -4]] )
* (see [[-.6 4 ] [1.6 -4]] * )
2nd MATRX (see highlight NAMES and highlight 1: )
Down cursor (to highlight 2:)
ENTER (see [A]-1 * [B] on screen)
ENTER
(see MATRIX [A] blink x2)
2 ENTER (puts in number of rows)
> > (moves cursor to blink)
2 ENTER (puts in the number of columns)
> > (moves cursor to highlight entry at 1,1)
1 ENTER (puts the value 1 in row i column 1)
(cursor moves to highlight entry at 1,2)
1 ENTER (puts the value 1 in row 1 column 2)
(cursor moves to highlight entry at 2,1)
0 . 4 ENTER (puts the value 0.4 in row 2 column 1)
(cursor moves to highlight entry at 2,2)
0 . 1 5 ENTER (puts the value 0.15 in row 2 column 2)
2nd MATRX (see highlight NAMES and highlight 1: )
ENTER (see [A] on new screen)
X-1 (see [A]-1)
ENTER (see [[-.6 4 ] [1.6 -4]] )
Continue item 12c on page 63 by entering the values for matrix B [[ 3000][ 840]]
2nd MATRX (see highlight NAMES and highlight 1: )
ENTER (see [A] on new screen)
X-1 (see [A]-1)
ENTER (see [[-.6 4 ] [1.6 -4]] )
* (see [[-.6 4 ] [1.6 -4]] * )
2nd MATRX (see highlight NAMES and highlight 1: )
Down cursor (to highlight 2:)
ENTER (see [A]-1 * [B] on screen)
ENTER
10-27 Class notes
Yesterday we found the determinant of a 2x2 matrix with entries [[a b][c d]]
as ad - bc.
Today we used the determinant in an expression to find the inverse matrix
(1/det) [[d -b][ -c a]]
If we have a 2x2 matrix of known values times a 2x1 matrix of unknowns X equal to a 2x1 matrix of known values,
A times [[a][ b]] = B where X = [[a][ b]] (see page 62, item 11)
[[ 1 1 ][0.4 0.15]] times [[a][ b]] = [[3000][ 840]] B = [[3000][ 840]]
we can isolate the matrix of unknowns by multiplying both sides of the equation by the inverse of the first matrix. This simplifies the left side, isolating the 2x1 matrix of variables.
The inverse of [[ 1 1 ][0.4 0.15]] is [[ -0.6 4][ 1.6 -4]] (see page 63, item 12b)
[[a][ b]] = inverse of A times B
[[a][ b]] = [[ -0.6 4][ 1.6 -4]] times [[3000][ 840]]
Homework for Friday 10-28: page 63, item 12c; Use matrix multiplication to find a and b in the 2x1 matrix of unknowns.
as ad - bc.
Today we used the determinant in an expression to find the inverse matrix
(1/det) [[d -b][ -c a]]
If we have a 2x2 matrix of known values times a 2x1 matrix of unknowns X equal to a 2x1 matrix of known values,
A times [[a][ b]] = B where X = [[a][ b]] (see page 62, item 11)
[[ 1 1 ][0.4 0.15]] times [[a][ b]] = [[3000][ 840]] B = [[3000][ 840]]
we can isolate the matrix of unknowns by multiplying both sides of the equation by the inverse of the first matrix. This simplifies the left side, isolating the 2x1 matrix of variables.
The inverse of [[ 1 1 ][0.4 0.15]] is [[ -0.6 4][ 1.6 -4]] (see page 63, item 12b)
[[a][ b]] = inverse of A times B
[[a][ b]] = [[ -0.6 4][ 1.6 -4]] times [[3000][ 840]]
Homework for Friday 10-28: page 63, item 12c; Use matrix multiplication to find a and b in the 2x1 matrix of unknowns.
Wednesday, October 26, 2011
Class notes 10-26 Matrix properties and the determinant
Homework for Thursday 10-27 is to read pages 57-60 in the textbook and to show your notes from today, Properties of matrix multiplication and the determinant.
Matrix addition is commutative.
Matrix subtraction is not commutative.
The identity element for matrix addition is a matrix of the same dimension with every entry = 0.
Matrix multiplication is not commutative; A*B does not equal B*A. Changing the order of the two matrices changes the resulting matrix.
For a square matrix, the identity element for mutiplication is a square matrix of the same dimension with 1's on the diagonal which goes from upper left to lower right, and 0's for all the other entries.
For a square matrix with dimension 2x2, with the elements arranged a and b in the top row and c and d in the bottom row, the determinant is a*d - b*c. The determinant will be useful in finding the inverse of certain matrices.
A matrix is a very compact way of representing an arrangement of values, relationships, and coefficients. Matrices can be manipulated and transformed using operations similar to the operations for real numbers.
Matrix addition is commutative.
Matrix subtraction is not commutative.
The identity element for matrix addition is a matrix of the same dimension with every entry = 0.
Matrix multiplication is not commutative; A*B does not equal B*A. Changing the order of the two matrices changes the resulting matrix.
For a square matrix, the identity element for mutiplication is a square matrix of the same dimension with 1's on the diagonal which goes from upper left to lower right, and 0's for all the other entries.
For a square matrix with dimension 2x2, with the elements arranged a and b in the top row and c and d in the bottom row, the determinant is a*d - b*c. The determinant will be useful in finding the inverse of certain matrices.
A matrix is a very compact way of representing an arrangement of values, relationships, and coefficients. Matrices can be manipulated and transformed using operations similar to the operations for real numbers.
10-26 Class notes
Homework for Thursday 10-27 is to read pages 57-60 in the textbook and to show your notes from today, Properties of matrix multiplication and the determinant.
Matrix addition is commutative.
Matrix subtraction is not commutative.
The identity element for matrix addition is a matrix of the same dimension with every entry = 0.
Matrix multiplication is not commutative; A*B does not equal B*A. Changing the order of the two matrices changes the resulting matrix.
For a square matrix, the identity element for mutiplication is a square matrix of the same dimension with 1's on the diagonal which goes from upper left to lower right, and 0's for all the other entries.
For a square matrix with dimension 2x2, with the elements arranged a and b in the top row and c and d in the bottom row, the determinant is a*d - b*c. The determinant will be useful in finding the inverse of certain matrices.
A matrix is a very compact way of representing an arrangement of values, relationships, and coefficients. Matrices can be manipulated and transformed using operations similar to the operations for real numbers.
Matrix addition is commutative.
Matrix subtraction is not commutative.
The identity element for matrix addition is a matrix of the same dimension with every entry = 0.
Matrix multiplication is not commutative; A*B does not equal B*A. Changing the order of the two matrices changes the resulting matrix.
For a square matrix, the identity element for mutiplication is a square matrix of the same dimension with 1's on the diagonal which goes from upper left to lower right, and 0's for all the other entries.
For a square matrix with dimension 2x2, with the elements arranged a and b in the top row and c and d in the bottom row, the determinant is a*d - b*c. The determinant will be useful in finding the inverse of certain matrices.
A matrix is a very compact way of representing an arrangement of values, relationships, and coefficients. Matrices can be manipulated and transformed using operations similar to the operations for real numbers.
Tuesday, October 18, 2011
EA2 Currency conversion 10-14 and 10-17
Page 47; items 1, 2, 3, 4
1. For E(D) = 0.64D - 5, the domain units are dollars and the range units are euros
1. For R(E) = 12.1E - 10, the domain units are euros and the range units are rand.
2. When D = 450, E(D) = 0.64(450 dollars) - 5 = 283 euros
When E = 283, R(E) = 12.1(283 euros) - 10 = 3414.30 rand
Compose the functions by putting one function inside the other.
3. R(E(D)) = R an outer function operating on E and inner function operating on D a variable.
R(E(D)) = 12.1(0.64D - 5) - 10
R(E(D)) = 7.744D - 70.5
When D = 450, R(E(450) = 7.744(450) - 70.5 = 3414.30 rand
1. For E(D) = 0.64D - 5, the domain units are dollars and the range units are euros
1. For R(E) = 12.1E - 10, the domain units are euros and the range units are rand.
2. When D = 450, E(D) = 0.64(450 dollars) - 5 = 283 euros
When E = 283, R(E) = 12.1(283 euros) - 10 = 3414.30 rand
Compose the functions by putting one function inside the other.
3. R(E(D)) = R an outer function operating on E and inner function operating on D a variable.
R(E(D)) = 12.1(0.64D - 5) - 10
R(E(D)) = 7.744D - 70.5
When D = 450, R(E(450) = 7.744(450) - 70.5 = 3414.30 rand
10-17 Adding matrices
Homework for 10-17 - Page 51 and 52; items Try these a, b, c, 8a, 9b, 9c
Scalar multiplication means multiplying number by a matrix.
4S, when S is the 3x3 matrix on page 49, means multiply each entry of S by 4.
This is similar to distributing the 4 over all the entries inside the grouping symbols, although this time it's brackets and not parentheses.
You can add or subtract matrices only if the dimensions are the same.
You can add a 2x2 to a 2x2. You can add a 3x2 to a 3x2.
You CANNOT add a 2x3 to a 3x2. You CANNOT add a 1x4 to a 2x3.
To add or subtract matrices, add or subtract corresponding entries. An entry is a corresponding entry if it has the same row and column position in its matrix as another entry has in its matrix.
A matrix is a simpler representation of a table of information.
The rows of a matrix have information about the labels on the left. Rows run left to right.
The columns of a matrix have information about the labels on top. Columns run up and down.
Scalar multiplication means multiplying number by a matrix.
4S, when S is the 3x3 matrix on page 49, means multiply each entry of S by 4.
This is similar to distributing the 4 over all the entries inside the grouping symbols, although this time it's brackets and not parentheses.
You can add or subtract matrices only if the dimensions are the same.
You can add a 2x2 to a 2x2. You can add a 3x2 to a 3x2.
You CANNOT add a 2x3 to a 3x2. You CANNOT add a 1x4 to a 2x3.
To add or subtract matrices, add or subtract corresponding entries. An entry is a corresponding entry if it has the same row and column position in its matrix as another entry has in its matrix.
A matrix is a simpler representation of a table of information.
The rows of a matrix have information about the labels on the left. Rows run left to right.
The columns of a matrix have information about the labels on top. Columns run up and down.
10-18 Multiplying matrices
Homework for 10-19 or after Career Fair
Page 56; items 16a, 16b, 16c and 1, 2, 3, 4, 5, 6, 7, 8 on the bottom
Page 53 in Springboard, item 10: The matrix multiplication example takes each person row and multiplies it by each wood column, using all 6 combinations of Monique and Shondra with Walnut, Maple and Cherry.
A 2x3 matrix with 2 rows and 3 columns can multiply a 3x2 matrix with 3 rows and 2 columns.
The result is a 3x3 matrix.
A 2x3 matrix and a 3x2 matrix can be multiplied because the # of columns in the first equals the # of rows in the second (2 = 2).
The result is a 3x3 matrix because that's the # of rows in the first and the # of columns in the second.
A 1x4 matrix with 1 rows and 4 columns can multiply a 4x2 matrix with 4 rows and 2 columns.
The result is a 1x2 matrix.
A 1x4 matrix and a 4x2 matrix can be multiplied because the # of columns in the first equals the # of rows in the second (4 = 4).
The result is a 1x2 matrix because that's the # of rows in the first and the # of columns in the second.
A 2x3 matrix with 2 rows and 3 columns CANNOT multiply a 2x3 matrix with 2 rows and 3 columns.
A 2x3 matrix and a 2x3 matrix CANNOT be multiplied because the # of columns in the first DOES NOT EQUAL the # of rows in the second (3 does not equal 2).
If you try to multiply a row by a column, the number of corresponding entries does not match.
Page 56; items 16a, 16b, 16c and 1, 2, 3, 4, 5, 6, 7, 8 on the bottom
Page 53 in Springboard, item 10: The matrix multiplication example takes each person row and multiplies it by each wood column, using all 6 combinations of Monique and Shondra with Walnut, Maple and Cherry.
A 2x3 matrix with 2 rows and 3 columns can multiply a 3x2 matrix with 3 rows and 2 columns.
The result is a 3x3 matrix.
A 2x3 matrix and a 3x2 matrix can be multiplied because the # of columns in the first equals the # of rows in the second (2 = 2).
The result is a 3x3 matrix because that's the # of rows in the first and the # of columns in the second.
A 1x4 matrix with 1 rows and 4 columns can multiply a 4x2 matrix with 4 rows and 2 columns.
The result is a 1x2 matrix.
A 1x4 matrix and a 4x2 matrix can be multiplied because the # of columns in the first equals the # of rows in the second (4 = 4).
The result is a 1x2 matrix because that's the # of rows in the first and the # of columns in the second.
A 2x3 matrix with 2 rows and 3 columns CANNOT multiply a 2x3 matrix with 2 rows and 3 columns.
A 2x3 matrix and a 2x3 matrix CANNOT be multiplied because the # of columns in the first DOES NOT EQUAL the # of rows in the second (3 does not equal 2).
If you try to multiply a row by a column, the number of corresponding entries does not match.
Friday, October 14, 2011
Assignment for Monday 10-17
Homework for Monday 10-19
Finish EA2 from the Springboard book page 47, items 1, 2, 3, 4
Hints: Units of the domain are what you are counting for the input. Units of the range are what you are counting for the output. To compose a function is to substitute a function E of D for the input E of an outside function R of E, so the outside function R of E operates on the inside function E of D. Simplifying the composite function R of E of D gives R of D, where its domain is the input of the inner function, and its range is the output of the outer function.
Also, the bell ringer was to plot four points A, B, C, D and draw a polygonal region.
Start with graph paper and prepare a grid which can fit up to 300 on the y-axis and up to 300 on the x-axis.
Use wide spacing so that one square corresponds to 10 units, which means 10 spaces correspond to 100 units.
The points have coordinates (x,y)
A: (0, 300)
B: ( ?, ? )
C: ( ?, ? )
D: (188, 0)
Point B is found by solving the system of equations
x + 2y = 600
3x + y = 600
Use combination of opposites and elimination to temporarily make one variable go away, solve for one variable, then use substitution to find the other variable.
Point C is found by solving the system of equations
9x + 3y = 1800
9x + y = 1700
Use substitution to temporarily make one variable go away, solve for one variable, then use substitution to find the other variable.
Plot points A, B, C, D, and join them with straight lines.
Refer to earlier blog posts on 9-19.
Finish EA2 from the Springboard book page 47, items 1, 2, 3, 4
Hints: Units of the domain are what you are counting for the input. Units of the range are what you are counting for the output. To compose a function is to substitute a function E of D for the input E of an outside function R of E, so the outside function R of E operates on the inside function E of D. Simplifying the composite function R of E of D gives R of D, where its domain is the input of the inner function, and its range is the output of the outer function.
Also, the bell ringer was to plot four points A, B, C, D and draw a polygonal region.
Start with graph paper and prepare a grid which can fit up to 300 on the y-axis and up to 300 on the x-axis.
Use wide spacing so that one square corresponds to 10 units, which means 10 spaces correspond to 100 units.
The points have coordinates (x,y)
A: (0, 300)
B: ( ?, ? )
C: ( ?, ? )
D: (188, 0)
Point B is found by solving the system of equations
x + 2y = 600
3x + y = 600
Use combination of opposites and elimination to temporarily make one variable go away, solve for one variable, then use substitution to find the other variable.
Point C is found by solving the system of equations
9x + 3y = 1800
9x + y = 1700
Use substitution to temporarily make one variable go away, solve for one variable, then use substitution to find the other variable.
Plot points A, B, C, D, and join them with straight lines.
Refer to earlier blog posts on 9-19.
Wednesday, October 12, 2011
10-12 Inverse functions class notes
When a function f operates on its inverse f^-1, you get back to the original input.
To create the inverse function of y = f(x), x and y trade places, then solve for y
To create the inverse function of y = f(x), x and y trade places, then solve for y
y = 2x - 4
x = 2y - 4
the inverse y = (x + 4)/2
Show that 2x - 4 and (x + 4)/2 are inverses by letting the original function operate on the inverse.
y = 2[(x + 4)/2] - 4
y = x, getting back to the original input
Example using x = 3
y = 2(3) - 4 = 2
this ouput 2 is the new input for the inverse y = (x + 4)/2
inverse y = (2 + 4)/2 = 6/3 = 2, the original input
Homework for 10-13: page 44 and 45, items 11, 12, 13, 14, 15
Thursday, October 6, 2011
10-6 Class notes
In practical expressions and formulas, the variable isolated on one side of the equal sign is usually the output or range variable. The other variable, mixed with operations and quantities, is usually the input or domain variable.
F(A) = Cost = 30 + 80A, where A is a domain value measured in acres. The range is measured in dollars.
G(C) = Area = C - 30, obtained by isolating A and solving for A. C is now a domain value measured 80
in dollars, and the range is the output measured in acres.
F and G are inverse functions. The domain and range traded places.
F(A) when A = 3/8 is F(3/8) = 30 + 80(3/8) = 30 + 30 = 60
G(F(A) uses the original function inside the inverse function.
Substitute 60 for F(A). Evaluate G(60).
G(60) = 60 - 30 = 30/80 = 3/8. Note that when 3/8 is the input to function F, 60 is the output.
80
When 60 is the input in the inverse function G, 3/8 is the output.
F(A) = Cost = 30 + 80A, where A is a domain value measured in acres. The range is measured in dollars.
G(C) = Area = C - 30, obtained by isolating A and solving for A. C is now a domain value measured 80
in dollars, and the range is the output measured in acres.
F and G are inverse functions. The domain and range traded places.
F(A) when A = 3/8 is F(3/8) = 30 + 80(3/8) = 30 + 30 = 60
G(F(A) uses the original function inside the inverse function.
Substitute 60 for F(A). Evaluate G(60).
G(60) = 60 - 30 = 30/80 = 3/8. Note that when 3/8 is the input to function F, 60 is the output.
80
When 60 is the input in the inverse function G, 3/8 is the output.
Wednesday, October 5, 2011
10-6 Homework quiz
Homework for Thursday 10-7.
page 40, items 1, 2, 3, 4, 5, 6, 7, 8, 9
If you did not finish the Wednesday 10-6 homework quiz, you must do the assigned problems on page 40. You can finish them in class with help from one of the students who handed in their himework quiz Wednesday.
If you finished the Wednesday homework quiz and handed it in, you do not have to write out the problems on page 40. You will be assigned to help a classmate who is finishing the quiz and the page 40 problems.
Homework quiz Wednesday 10-6:
page 32; 3c, 4a, 5b
page 37; 16a, 16b
page 38; 17a, 17b
Concepts: When evaluating a function of a function, use the variable or value with the inside function first. Run the variable through the operations or the table for the inside function. Then run that result through the operations or table of the outside function.
page 40, items 1, 2, 3, 4, 5, 6, 7, 8, 9
If you did not finish the Wednesday 10-6 homework quiz, you must do the assigned problems on page 40. You can finish them in class with help from one of the students who handed in their himework quiz Wednesday.
If you finished the Wednesday homework quiz and handed it in, you do not have to write out the problems on page 40. You will be assigned to help a classmate who is finishing the quiz and the page 40 problems.
Homework quiz Wednesday 10-6:
page 32; 3c, 4a, 5b
page 37; 16a, 16b
page 38; 17a, 17b
Concepts: When evaluating a function of a function, use the variable or value with the inside function first. Run the variable through the operations or the table for the inside function. Then run that result through the operations or table of the outside function.
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