Matrices
Identity Matrix
-It is always a square matrix
-Square Matrix: same # of rows and columns
-Square Matrix: same # of rows and columns
Adding/Subtracting Matrices
-Add/subtract elements in the same positions
-To add/subtract: matrices must have same dimensions (ex. 2 by 2)
-To add/subtract: matrices must have same dimensions (ex. 2 by 2)
Multiplying Matrices
In order to multiply matrices, the number of columns in the first matrix needs to equal the number of rows in the second matrix
Scalar Multiplication
In this example: The scalar is 2. The scalar is multiplied by every element in the matrix
To find the inverse of a Matrix in the Calculator
Step 1: Press 2nd on the calculator. Then, press matrix.
Step 2: Go to edit and pick a letter (ex. [A], [B], [C], etc.) where you will store your martix
Step 3: Make sure the matrix has the same amount of rows and columns as the matrix you are trying to find the inverse of
Step 4: Put the same values that are found in your matrix that you are trying to find the inverse of in the correct positions
Step 5: Exit out of the matrix
Step 6: Press 2nd and Matrix
Step 7: Select the letter where you stored your matrix
Step 8: Press x^-1 button on your calculator and press enter
Step 9: Answer
Step 2: Go to edit and pick a letter (ex. [A], [B], [C], etc.) where you will store your martix
Step 3: Make sure the matrix has the same amount of rows and columns as the matrix you are trying to find the inverse of
Step 4: Put the same values that are found in your matrix that you are trying to find the inverse of in the correct positions
Step 5: Exit out of the matrix
Step 6: Press 2nd and Matrix
Step 7: Select the letter where you stored your matrix
Step 8: Press x^-1 button on your calculator and press enter
Step 9: Answer
Finding Determinants
-Determinants are only found in a square matrix
Step 1: Multiply the elements diagonally (ex. 5*2 and 8*4)
(If there are more than one "up" answer, add the answers together to form one answer. If there are more than one "down" answer, add the answers together to form one answer)
Step 2: Subtract the downs from the ups (5*2)-(8*4)
Step 3: Answer= (-22)
(If there are more than one "up" answer, add the answers together to form one answer. If there are more than one "down" answer, add the answers together to form one answer)
Step 2: Subtract the downs from the ups (5*2)-(8*4)
Step 3: Answer= (-22)
Finding Determinants with a Calculator
Step 1: Press 2nd and Matrix---> Go to Edit
Step 2: Enter in the values in the matrix
Step 3: Exit out of the editing
Step 4: Press 2nd and Matrix
Step 5: Press math and det()
Step 6: Press 2nd and Matrix--->Choose the matrix (ex. [A], [B], [C]) the values are under
Step 7: Press enter and your answer will show up
Step 2: Enter in the values in the matrix
Step 3: Exit out of the editing
Step 4: Press 2nd and Matrix
Step 5: Press math and det()
Step 6: Press 2nd and Matrix--->Choose the matrix (ex. [A], [B], [C]) the values are under
Step 7: Press enter and your answer will show up
Expansion by Minors
Step 1: Copy your determinant so there are the same amount of determinants as the amount of columns in one determinant (ex. 3 determinants= 3 columns in one of the determinants)
Step 2: Cross out the first row in each determinant and cross out each column between the determinants present (ex. The first determinant crosses out the first column. The second determinant crosses out the second column. The third determinant crosses out the third column.)
Step 3: Use the values that are not crossed out to make new determinants (Put the element that was crossed out twice next to the new determinant)
Step 4: Solve the determinants and multiply the answers by the element that was crossed out twice in the old determinants
Step 5: Assign a + (addition sign to the first answer), assign a - (subtraction sign to the second answer), and continue to switch the signs depending on the number of answers you have after solving the determinants
Step 6: Do what the signs say--> subtract when there is a - and add when there is a +
Step 7: After you add all the answers together---> your final answer will be shown
Step 2: Cross out the first row in each determinant and cross out each column between the determinants present (ex. The first determinant crosses out the first column. The second determinant crosses out the second column. The third determinant crosses out the third column.)
Step 3: Use the values that are not crossed out to make new determinants (Put the element that was crossed out twice next to the new determinant)
Step 4: Solve the determinants and multiply the answers by the element that was crossed out twice in the old determinants
Step 5: Assign a + (addition sign to the first answer), assign a - (subtraction sign to the second answer), and continue to switch the signs depending on the number of answers you have after solving the determinants
Step 6: Do what the signs say--> subtract when there is a - and add when there is a +
Step 7: After you add all the answers together---> your final answer will be shown
Cramer's Rule
For the equation above:
Step 1: Take the coefficients of the x and y in the equations and put them into a determinant (Name it D)
Step 2: Take the coefficients of the x and y in the equations and put them into a determinant, but replace the x coefficients with the answers of the equations (ex. 4 and 8) (Name it Dx)
Step 3: Take the coefficients of the x and y in the equations and put them into a determinant, but replace the y coefficients with the answers of the equations (ex. 4 and 8)
(Name it Dy)
Step 4: Solve the determinants (Remember: Subtract downs from ups after you multiply diagonally)
Step 5: Divide Dx and Dy by D
Step 6: Answer
Step 1: Take the coefficients of the x and y in the equations and put them into a determinant (Name it D)
Step 2: Take the coefficients of the x and y in the equations and put them into a determinant, but replace the x coefficients with the answers of the equations (ex. 4 and 8) (Name it Dx)
Step 3: Take the coefficients of the x and y in the equations and put them into a determinant, but replace the y coefficients with the answers of the equations (ex. 4 and 8)
(Name it Dy)
Step 4: Solve the determinants (Remember: Subtract downs from ups after you multiply diagonally)
Step 5: Divide Dx and Dy by D
Step 6: Answer
Systems of Equations
If you have 3 equations, you NEED 3 equations. (# variables must equal # of equations)
If you get a matrix row with all zeros--- there are infinite solutions
If you get a matrix row with all zeros, but a ending value--- there are no solutions
If you get a matrix row with all zeros--- there are infinite solutions
If you get a matrix row with all zeros, but a ending value--- there are no solutions
Partial Fractions
Questions:
#1:Do the Following with A and B Matrices (the first two matrices in the picture below with a red A and B)
A. A+B B. A-B C. 3A D. 2A-3B
Answers Above.
A. Addition
B. Subtraction
C. Scalar--Multiply every element by 3
D. Subtraction after multiply every element in the A matrix by 2 and multiply every element in the B matrix by 3
A. Addition
B. Subtraction
C. Scalar--Multiply every element by 3
D. Subtraction after multiply every element in the A matrix by 2 and multiply every element in the B matrix by 3
#2:Find A*B without a calculator and B*A with a calculator (A and B matrices are the top matrices in the picture below)
Answer Above.
#3:Use Cramer's rule to solve: x-2y=21 and 3x+y=7
Step 1: Take the coefficients of the x and y in the equations and put them into a determinant (Name it D)
--->Take the coefficients of the x and y in the equations and put them into a determinant, but replace the x coefficients with the answers of the equations (ex. 4 and 8) (Name it Dx)
Step 2: Take the coefficients of the x and y in the equations and put them into a determinant, but replace the y coefficients with the answers of the equations (ex. 4 and 8)
(Name it Dy)
-->Solve the determinants (Remember: Subtract downs from ups after you multiply diagonally)
Step 3: Divide Dx and Dy by D
Step 4: Answer (5,-8)
--->Take the coefficients of the x and y in the equations and put them into a determinant, but replace the x coefficients with the answers of the equations (ex. 4 and 8) (Name it Dx)
Step 2: Take the coefficients of the x and y in the equations and put them into a determinant, but replace the y coefficients with the answers of the equations (ex. 4 and 8)
(Name it Dy)
-->Solve the determinants (Remember: Subtract downs from ups after you multiply diagonally)
Step 3: Divide Dx and Dy by D
Step 4: Answer (5,-8)
Common Core:
Sally ordered 200 flowers for her friend. She ordered violets at $1.50 each, roses at $5.75 each, and daisies at $2.60 each. She ordered mostly violets, and 20 fewer roses than daisies. The total order came to $589.50. How many of each type of flower was ordered?
Answers:
Using Matrices!!!!!!!
Answers:
Using Matrices!!!!!!!
Answer above.
Lets take the Long Way!!!: Solve the system of equations by hand!!!
Step 1: Form 3 equations
Step 2: Multiply the 1st equation by -1.5 (to eliminate the A variable)
Step 2: Multiply the 1st equation by -1.5 (to eliminate the A variable)
Step 3:Add the 1st equation and 2nd equation together ( The variable A cancels out)
Step 4:Take the 3rd equation and solve for C
Step 5: Substitute the C from the 3rd equation into the new equation (1st equation and 2nd equation added together=new equation)
Step 6: Solve for B (B=50)
Step 7: Substitute the value of B into the 3rd equation to find out the value of C (C=70)
Step 4:Take the 3rd equation and solve for C
Step 5: Substitute the C from the 3rd equation into the new equation (1st equation and 2nd equation added together=new equation)
Step 6: Solve for B (B=50)
Step 7: Substitute the value of B into the 3rd equation to find out the value of C (C=70)
Step 8: Substitute the values of C and B into the (original) first equation and solve for A
Step 9: Answer
Step 9: Answer