IN CLASS
We define matrix operations like addition , subtraction and multiplication so
that we can manipulate
and solve systems of linear equations that are in matrix form like the following
"just like" we solve equations of one variable like 2x =
7.
MATRIX ADDITION AND SUBTRACTION
We define matrix addition so that
Ax + Bx = (A + B)x
is true for systems of equations in matrix form just like it is for equations of
one variable. For 2x2
matrices, for example, we define
Generalizing on this example we define the sum of two
matrices with the same dimensions to be
the sum of the corresponding elements. In the same way we define the difference
between two
matrices A - B of the same dimensions to be the difference of the corresponding
elements.
1. Matrix addition and subtraction
a. What happens when we add and subtract matrices
2. When matrix addition and subtraction are not defined
a. What has to be satisfied before we can add or subtract
two matrices
3. Matrices and numbers
a. What happens when we multiply a scalar times a matrix
b. What happens when we add a scalar to a matrix
MATRIX MULTIPLICATION
Analogously to the case of matrix addition and subtraction we define matrix
multiplication so that
y = Aw and w = Bx implies that y = ABx
for systems of equations in matrix form just like it does for equations of one
variable. For example
suppose
Then by direct substitution we have
Which in matrrix form is equal to
We therefore define the matrix product AB to be the
coefficient of the vector x in our equation
y = ABx as follows
This looks complicated at first but in fact it's not with
= sum of the products when the elements in the first row of A are multiplied
by the first column of B and then added together
= sum of the products when the elements in the first row of A are multiplied
by the second column of B and then added together
and so on. Note that the product AB can be calculated only if A has the same
number of columns
as B has rows.
4. Multiplying square matrices
a. What happens when we multiply two matrices
b. What has to be satisfied before we can multiply two matrices
5. The identity matrix I
a. What happens when we multiply a matrix A by the
identity matrix
6. Some simple matrix factoring
7. The distributive law
a. How is the distributive law for matrices like the
distributive law in regular algebra
8. More matrix multiplication
a. How does the dimension of A have to be related to the
dimension of B for the product
A*B to be defined
b. How is the condition for A*B to exist different from the condition for A+B to
exist
INVERSE MATRICES AND MATRIX DIVISION
9. The inverse matrix
a. What does inv(A) do. How can you tell
b. Can we take the inverse of a nonsquare matrix
10. Solving Ax = b by calculating x = A-1b
a. How did we use the inverse of A to calculate x
11. Parallel lines
a. What happened when we tried to calculate the
intersection of two parallel lines
12. Faster solving of Ax = b
a. Why is x = A \ b faster than x = A-1b
