Math 220 - February 10, 1999

Discussion

• The n \times n identity matrix 1_{n} is the matrix

  (

  1	0	0	...	0
  0	1	0	...	0
  :	:	:	:	:
  0	0	0	...	1

  ) .

  The identity matrix has the property that multiplying another matrix by it, whenever the sizes are correct, leaves the other matrix as the result.

• A square matrix A is invertible if there is another matrix B of the same size such that

  A B = 1_{n} and B A = 1_{n} .

  If this is the case, there can be only one such B, and one writes A^{-1} for the unique such B.

• If A is an invertible matrix, then the system of linear equations A x = b has only one solution x. This solution is given by x = A^{-1} b.

• An invertible matrix A may be inverted by performing row operations on the multiply augmented matrix

  (A 1_{n})

  to maneuver it to reduced row echelon form, which (if the matrix is invertible) will be

  (1_{n} A^{-1}) .

Assignment for Friday, February 12

1. Invert the matrix

   (

   {1}/{SQRT{2}}	{1}/{SQRT{2}}
   -{1}/{SQRT{2}}	{1}/{SQRT{2}}

   ) .

2. Invert the matrices

   A = (

   1	1
   0	1

   ) and B = (

   0	1
   -1	0

   ) .

3. Invert the matrices

   A = (

   cos t	- sin t
   sin t	cos t

   ) and B = (

   cos t	sin t
   sin t	- cos t

   ) .

4. Solve the 3 \times 3 system of linear equations represented by

   U x = b

   when the column b is arbitrary and U is the matrix

   (

   0	0	1
   0	1	0
   1	0	0

   ) .