Introduction to matrices

4.1 and 4.2 Day 1

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

That’s RC Cola!

Would you like to solve THis….BY hand

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

- ???

Multiply a Matrix by a scalar

• For the following matrix A, find 2A and –1A.

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:

=

Multiply If Possible

=

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?

Applet

• http://www.mathresource.iitb.ac.in/linear%20algebra/example2.0.1/index.html