MATRICES
Part 1
The Identity Matrix
Sizes of Matrices
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To show how many rows and columns a matrix has we often write rows×columns.
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Part 2- What can we do with Matrices?
Multiplication within Matrices
How do we multiply matrices with each other ?
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Example:
When multiplying a matrix by another matrix, then it is going to be more complex. We need to multiply row with column, row with column (1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11 = 58
After you find one answer, then you continue doing the next row times the next column, and so forth. (1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12 = 64
Continue the bottom row with the column: (4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11 = 139 (4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12 = 154 and you will get this answer. |
IMPORTANT REMINDER WHEN MULTIPLYING A MATRIX BY ANOTHER MATRIX:
The number of columns of the first matrix must be equal to the number of rows of the second matrix.
The number of columns of the first matrix must be equal to the number of rows of the second matrix.
Adding/Subtracting Matrices
Part 3-Determinant of a Matrix
What is a determinant?
How does it look? How do we find it? |
The determinant of a matrix is a number that can be calculated from a square matrix.
The symbol for a Determinant Example: |A| means the determinant of the matrix A FINDING THE DETERMINANT
Example: (3x3 Matrix) Then you multiplyThe determinant is:
[(a)(e)(i)]+[(d)(h)(c)]+[(b)(f)(g)] - [(g)(e)(c)]-[(d)(b)(i)]-[(f)(h)(a)] |
Solving with Expansion by Minors
What is the purpose of a determinant?
Challenging Example:
An outbreak of Chicken Pox hit the local public schools. Approximately 15% of the male and female juniors and 25% of the male and female seniors are currently healthy, 35% of the male and female juniors and 30% of the male and female seniors are currently sick, and 50% of the male and female juniors and 45% of the male and female seniors are carriers of Chicken Pox.
There are 100 male juniors, 80 male seniors, 120 female juniors, and 100 female seniors.
Find out how many males and how many females (don’t need to divide by class) are healthy, sick, and carriers.
Step 1-Set up a matrix
Since we have males and females that are both jinors and seniors , we need to set up a matrix with the overal number of males and femals per class.
Step 2- set up a matrix for the percentage of healthy, sick, and chicken pox carriers, that are males and females
Step 3- Multiply the two matrices
Step 4- Answer: There will be 35 healthy males, 59 sick males, and 86 carrier males, 43 healthy females, 72 sick females, and 95 carrier females.
An outbreak of Chicken Pox hit the local public schools. Approximately 15% of the male and female juniors and 25% of the male and female seniors are currently healthy, 35% of the male and female juniors and 30% of the male and female seniors are currently sick, and 50% of the male and female juniors and 45% of the male and female seniors are carriers of Chicken Pox.
There are 100 male juniors, 80 male seniors, 120 female juniors, and 100 female seniors.
Find out how many males and how many females (don’t need to divide by class) are healthy, sick, and carriers.
Step 1-Set up a matrix
Since we have males and females that are both jinors and seniors , we need to set up a matrix with the overal number of males and femals per class.
Step 2- set up a matrix for the percentage of healthy, sick, and chicken pox carriers, that are males and females
Step 3- Multiply the two matrices
Step 4- Answer: There will be 35 healthy males, 59 sick males, and 86 carrier males, 43 healthy females, 72 sick females, and 95 carrier females.