This is the second of two weeks devoted to solving systems of linear equations. It begins with two sections on matrices

**Chapter 4-4****Chapter 4-5**

after which we cover one more method for solving linear equations:

**Chapter 4-6**: The Inverse Matrix method for solving systems of linear equations

We will skip section 4-7.

An *mxn matrix* is a rectangular array of *mn* numbers,
arranged in *m* rows of *n* numbers each. Matrices can be added,
subtracted, and multiplied by numbers or by other matrices, as long as the
matrix sizes are compatible.

"Compatible" means different things for different operations: to add or subtract
two matrices, they have to be of the same size. To multiply two matrices, their "inner
dimensions" have to agree. If *A* is an *mxn* matrix,
and *B* is an *nxp* matrix, and you put
the dimensions side by side: *(mxn)(nxp)*,
the "inner dimension"
is *n*, and the "outer dimensions" are *m* and *p*. The inner
dimension disappears, and the result is of size *mxp* (the
outer dimensions).

We use the notation 0 for the **zero matrix**, which is
a matrix full of zeros, and talk about ** the** zero matrix, even
though technically there are lots of different zero matrices: a zero matrix
of size

If the product *AB* exists, the product *BA* does not necessarily
also exist. Even if it does, it does not have to be of the same size: if *A* is
of size *2x3*, and *B *is of size *3x2*,
then *AB* has size *2x2*, and *BA* has
size *3x3*. Even if *A* and *B* are
square matrices of the same size, so that both *AB* and *BA* are
also of the same size, they don't have to be the same: matrix multiplication
is not commutative.

Another property of matrix multiplication which is different from multiplying
numbers is that it is possible to have two non-zero matrices *A*, *B* with *AB=*0.

Except for the properties mentioned in the last two paragraphs, matrix arithmetic
satisfies all the usual properties of standard arithmetic. For example, *A(B+C)
= AB + AC*, *A(BC) = (AB)C*, and so on.

The **identity matrix** of size *nxn* is
the matrix* I* which has 1 on the principal diagonal, 0 everywhere else.
It has the property that *IA = A* and *AI = A*, so *I* corresponds
to the number 1. As with the zero matrix, there are really lots of different
identity matrices, one for each (square) size.

If you can find a pair of matrices *A*, *B* with *AB=I* and *BA=I*,
then *B* is called the **inverse of A**, written

Inverse matrices can be used to solve matrix equations. If *AX=B*, then *X=A ^{-1}B*.
Pay attention to the order:

A more complicated example is this: Solve *AX+X=B*.

Solution: *AX+X
= AX+IX = (A+I)X = B*, so *B=(A+I) ^{-1}B*.

Read the textbook and do the homework assignment HW 5.

Last Updated: Wednesday, August 5, 2015