Matrix Operations
Part 6: Summary
The questions in this summary
are about arbitrary square matrices A, B, C, of the
same size.
- Are the products AB
and BA always the same? If not, is there some special family of
square matrices for which they are always the same?
- Are the products A(BC)
and (AB)C always the same? If not, is there some special family
of square matrices for which they are always the same?
- Do the expressions A(B+C)
and AB+AC always represent the same matrix? If not, is there some
special family of square matrices for which they are always the same?
- Suppose x is a column
vector with the same number of rows as A. How can the vector Ax
be written as a linear combination of the columns of A?
- Complete the following
sentence: The identity matrix behaves multiplicatively much like the
real number ...
- Complete the following
sentence: The zero matrix behaves multiplicatively much like the real
number ...
- Are zero and identity matrices
also diagonal matrices? Explain.
- Which diagonal matrices
are invertible? Explain.
- How are determinants related
to invertibility? How is the determinant of an invertible matrix A
related to the determinant of A-1?
- How is the determinant
of a product AB related to the determinants of A and B?
- How is the determinant
of a sum A+B related to the determinants of A and B?
- What are the possible numbers
of solutions of a system of linear equations, Ax = b? If A
is an invertible matrix, how does your answer to this question change?
