- Write down the 3 by 3 matrices with entries
- Compute the products and and
For the matrices compute , and and and and
Find examples of 2 by 2 matrices with for with
Solve by elimination and back-substituion:
Factor the preceding matrices in to or .
- There are sixteen 2 by 2 matrices whose entries are 1’s and 0’s How many are invertible?
- (Much harder!) If you put 1’s and 0’s at random into the entires of a 10 by 10 matrix its more likely to be invertible or singular?
There are sixteen matrices 2 by 2 whose entries are 1’s and -1’s. How many are invertible?
How are the rows of related to the rows of if
Write down a 2 by 2 system with infinitely many solutions.
Find inverses if they exist by inspection or by Gauss-Jordan
If is 2 by 2 and it adds the first equation to the second what are and and ?
True or fals, with reason if true or counterexample if false:
- If is invertible an its rows are in reverse order in then is invertible.
- If and are summetric then is symmetric.
- If and are invertible then is invertible
- Every nonsingular matrix can be factored in to the product of a lower triangular and an upper triangular .
Solve by solving the triangular systems and . What part of have you found with this particular ?
If possible find 3 by 3 matrices B such that
- for every
- for every
- has the first and last rows of reversed
- has the first and last columns of reversed
Find the value for in the following by inverse:
for which values of does have no solution, one solution or infinitely many solutions?
Find the symmetric factorization of
Suppose is the 4 by 4 identity matrix except for a vector in the column 2;
- Factor in to assuming
- Find , which has the same form as .
Solve by elimination or show that there is no solution:
The by permuation matrices are an important example of a “group”. If you multiply them you stay inside the group; they have inverses in the group; the identity is in the group; and the laws is true – because it is true for all matrices.
- How may members belong to the group of 4 by 4 matrices and by permuation matrices?
- Find a power k so that all 3 by 3 permutation matrices satisfy
Describe the rows of and columns of if .
- If is invertible wha is the inverse of
- If A is also symmetric what is the transpose of
- Illustrate both formulas when .
By experiment with and find
Starting with a first plane find the equations for
- the parallel plane through the origin
- a second plane that also contains the points and
- a third plane that meets the first and second in the point
What multiple of row 2 is substracted from row 3 in forward elimination of How do you know (without multiplying those factors) that is invertible, symmetric and tridiagonal? What are the pivots?
- What vector will make column 1 + 2(column 3), for all 3 by 3 matrices ?
- Construct a matrix which has column 1 + 2(column 3) = 0. Check that A is singular (fewer than 3 pivots) and explain why that must happen.
True or false, with reason if true and counter example if false:
- If (upper triangulars ’s with nonzero diagonal, lower triangular ’s with unit diagonal) then and . The factorization is unique.
- If then
- If all diagonal entries of are zero then is singular.
By experiment or Gauss-Jordan Method compute
Write down the 2 by 2 matrices which
- reverse the direction for every vector
- project every vector onto the -axis
- turn every vector counterclowise through
- reflect every vector through the line