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.
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