Math 110 - 9/19/02

More on Matrix Multiplication

A× B is not always equal to B× A. It may not even be a permissible operation in one direction:

A [2 by 4 matrix]× [4 by 3 matrix] can be done, and results in a [2 by 3 matrix].

But a [4 by 3 matrix] × [2 by 4 matrix] cannot be done.

Section 2.6 - Inverse of a matrix

When we need to divide by a matrix - instead, we multiply by its inverse.

Identity matrix I has 1's on diagonal, all others 0.

 

I2 =      I3 =

A matrix times its inverse is an identity matrix:

AA-1 = I . Only square matrices have inverses.

Try problem 1, p.105
Example 2, p.99, shows finding inverse using algebra.
Better way: p.102 blue box and Example 5.

Try problem 11. Start:

Some square matrices have no inverse. See Example 6, p. 103. Then try problem #23, p.107.

More practice: #7.

Where all of this is going:
We can write a set of equations as AX = B, where A is a matrix, X is the set of unknown variables, maybe x1, x2, x3, and B is the last column of the augmented matrix, or the right hand side.

If A and B were constant single numbers, we would solve AX = B by dividing each side by A, so X=B/A. Dividing by A is the same as multiplying by A's inverse.
So we solve the whole set of equations by using the inverse of the matrix A:

A-1AX = X = A-1B

We find the inverse of A to solve for all the values x1, x2, x3,...

See Example 7. Then try #39