Find the inverses (no special system required) of
- Find the inverse of the permuation matrices
- Explain why for permutations is always the same as , by using showing that the ’s are in the right places to give .
From find a formula . Do the same for .
- If A is invertible and prove quickly that .
- If , find an example with but .
If the inverse of is show that the inverse of is . (Thus whenever is invertible)
Use the Gauss-Jordan method to invert
Find three 2 by 2 matrices other than and that are their own inverse .
Show that , has no inverse by trying to solve
When elimination fails for singular matrices like show that cannot be invertible. The third row of , multiplying A, should give the third row of . Why is this impossible?
Find the inverses (in any legal way) of:
give examples of and such that:
- is not invertible although and are invertible
- is invertible although and are not invertible
- all of and are invertible In the last case use to show that is also invertible — and find a formula for its inverse.
Which properties of a matrix are preserved by its inverse (assuming exists)? (1) is trinagular, (2) A is Symetric, (3) A is tridiagonal, (4) all entries are whole numbers, (5) all entries are fraction (including whole numbers like )
If and compute .
(Important) Prove that even for rectangular matrices and are always symmetric. Show by example that they may not be equal even for square matrices.
Show that for any wquare matrix is always symmetric and is aways skew-smmetric — which means that . Find these matrices when , and write as the sum of a symmetric matrix and a skewsymetrix matrix.
- How many entries can be chosen indeppendently, in a symmetric matrix of order ?
- Hoe many entries can be chosen independently in a skew-symmetric matrix of order ?
- If A = LDU, which 1’s on the diagonals of and , what is the corresponding factorization of ? Note that and (squres matrices with no row exchanges share the same pivots.)
- What triangula system will give solution to
If and , prove that and , If is invertible the factorization is unique.
- Derive the equation for and explain why one side is lower triangular and the other side is upper triangular.
- Compare the main diagonals in the eauation, and then compare the offdiagonals.
Under what conditions on its entries is invertible if:
If the 3 by 3 matrix has row 1 + row 2 = row 3, show that it is impossible to solve . Can be invertible?
Compute the symmetric factorization of
Find the inverse of
If and are suqre matrices, show that is invertible if is invertible. Start from .