Do Now
Grab some slides from the front and solve this:Example: The local shop sells 3 types of pies.•Beef pies cost $3 each•Chicken pies cost $4 each•Vegetable pies cost $2 eachAnd this is how many they sold in 4 days:
Calculate the sales for each day.
Where we are and where we are going
• TODAY (10/12): 4.1 and 4.2 (Introduction to Matrices)
• THURSDAY (10/13): 4.2 (Applying Matrix Multiplication)
• FRIDAY (10/14) and MONDAY (10/17): 4.3 (Determinants and Cramer’s Rule)
• MONDAY (10/17): Princess Project Due!
• TUESDAY (10/18): 4.1-4.3 Performance Assessment
Essential vocabulary
• Matrix: a rectangular array of numeric or algebraic quantities subject to mathematical operations
For example, A is a matrix. The dimensions of A are 3x4 (RC Cola!).The numbers are called entries.
The Matrix Zoo!
• Row matrixA matrix with only 1 row
• Column matrixA matrix with only 1 column
• Square matrixA matrix with the samenumber of rows and columns
More Animals!
• Zero matrixA matrix with all entries equal to 0
• Identity matrix (of size n)An nxn matrix with ones on the main diagonal and
zeroes elsewhere
Adding and Subtracting Matrix elements
• Only if they have the same dimensions!
• Example:
How about
- ???
Matrix Equations
4*(-2x)=-48, so x=6 and 4(2y+6)=6, so 8y+24=6, so y=-9/4
2y+5=-5, so y=-5 and -6+x=-7, so x=-1
Matrix Multiplication: Size matters!
• If A is an m x n matrix, and B is an n x p matrix, then the product AB is an m x p matrix
• The number of columns of A must equal the number of rows of B for matrix multiplication to be defined.
• http://www.mathresource.iitb.ac.in/linear%20algebra/example2.0.1/index.html
•
• Example:
=
Refer back to “Do Now”
• Now think about this ... the value of sales for Monday is calculated this way:
• Beef pie value + Chicken pie value + Vegetable pie value
• $3×13 + $4×8 + $2×6 = $83• So it is, in fact, the "dot product" of prices and how
many were sold:• ($3, $4, $2) • (13, 8, 6) = $3×13 + $4×8 + $2×6 =
$83• We match the price to how many sold, multiply
them, then sum the result.
Graphing Calculator (TI-83): Multiplying Matrices
• Let’s verify this one:
• Hit “MATRX” (TI-84, 2nd
X^(-1)) • Scroll to Edit • Press 1• Enter dimensions of first matrix (3X2), Press Enter• Put in your entries, hit Enter after each entry• Hit “MATRX”, “Edit” again. This time, select [B] (2X3)• Now hit “2nd-MODE” to get to main screen and hit
“MATRX”, “1”, “X (times)”, “MATRX”, “2”, “ENTER
Hmmmm…..
• Using your calculator and the same two matrices, now compute B*A. What do you notice?
• Try entering the 3X3 identity matrix for A and multiplying it by any matrix B with the same dimensions. What do you notice?