ekiim's blog
1. Find the inverses (no special system required) of$A_{1}=\left[ \begin{array}{ll}{0} & {2} \\ {3} & {0}\end{array}\right], \quad A_{2}=\left[ \begin{array}{ll}{2} & {0} \\ {4} & {2}\end{array}\right], \quad A_{3}=\left[ \begin{array}{rr}{\cos \theta} & {-\sin \theta} \\ {\sin \theta} & {\cos \theta}\end{array}\right]$

1. Find the inverse of the permuation matrices $P=\left[ \begin{array}{lll}{0} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {0}\end{array}\right] \quad \text { and } \quad P=\left[ \begin{array}{lll}{0} & {0} & {1} \\ {1} & {0} & {0} \\ {0} & {1} & {0}\end{array}\right]$
2. Explain why for permutations $P^{-1}$ is always the same as $P^T$, by using showing that the $1$’s are in the right places to give $PP^T = I$.
2. From $AB=C$ find a formula $A^{-1}$. Do the same for $PA =LU$.

1. If A is invertible and $AB =AC$ prove quickly that $B=C$.
2. If $A = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$, find an example with $AB= AC$ but $B \neq C$.
3. If the inverse of $A^2$ is $B$ show that the inverse of $A$ is $AB$. (Thus whenever $A^2$ is invertible)

4. Use the Gauss-Jordan method to invert $A_{1}=\left[ \begin{array}{lll}{1} & {0} & {0} \\ {1} & {1} & {1} \\ {0} & {0} & {1}\end{array}\right], \quad A_{2}=\left[ \begin{array}{rrr}{2} & {-1} & {0} \\ {-1} & {2} & {-1} \\ {0} & {-1} & {2}\end{array}\right], \quad A_{3}=\left[ \begin{array}{ccc}{0} & {0} & {1} \\ {0} & {1} & {1} \\ {1} & {1} & {1}\end{array}\right]$

5. Find three 2 by 2 matrices other than $A=I$ and $A = -I$ that are their own inverse $A^2 = I$.

6. Show that $\begin{bmatrix} 1 & 1\\3 & 3\end{bmatrix}$, has no inverse by trying to solve $\left[ \begin{array}{ll}{1} & {1} \\ {3} & {3}\end{array}\right] \left[ \begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]=\left[ \begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]$

7. When elimination fails for singular matrices like $A=\left[ \begin{array}{llll}{2} & {1} & {4} & {6} \\ {0} & {3} & {8} & {5} \\ {0} & {0} & {0} & {7} \\ {0} & {0} & {0} & {9}\end{array}\right]$ show that $A$ cannot be invertible. The third row of $A^{-1}$, multiplying A, should give the third row of $A^{-1}A = I$. Why is this impossible?

8. Find the inverses (in any legal way) of:$A_{1}=\left[ \begin{array}{llll}{0} & {0} & {0} & {1} \\ {0} & {0} & {2} & {0} \\ {0} & {3} & {0} & {0} \\ {4} & {0} & {0} & {0}\end{array}\right], \quad A_{2}=\left[ \begin{array}{rrrr}{1} & {0} & {0} & {0} \\ {-\frac{1}{2}} & {1} & {0} & {0} \\ {0} & {-\frac{2}{3}} & {1} & {0} \\ {0} & {0} & {-\frac{3}{4}} & {1}\end{array}\right], \quad A_{3}=\left[ \begin{array}{cccc}{a} & {b} & {0} & {0} \\ {c} & {d} & {0} & {0} \\ {0} & {0} & {a} & {b} \\ {0} & {0} & {c} & {d}\end{array}\right]$

9. give examples of $A$ and $B$ such that:

1. $A+B$ is not invertible although $A$ and $B$ are invertible
2. $A + B$ is invertible although $A$ and $B$ are not invertible
3. all of $A, B$ and $A+B$ are invertible In the last case use $A^{-1}(A+B)B^{-1} = B^{-1} + A^{-1}$ to show that $B^{-1} + A^{-1}$ is also invertible — and find a formula for its inverse.
10. Which properties of a matrix $A$ are preserved by its inverse (assuming $A^{-1}$ exists)? (1) $A$ 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 $\frac31$)

11. If $A = \begin{bmatrix}3 \\ 1\end{bmatrix}$ and $B = \begin{bmatrix}2 \\ 2\end{bmatrix}$ compute $A^TB,B^TA, AB^T, BA^T$.

12. (Important) Prove that even for rectangular matrices $AA^T$ and $A^TA$ are always symmetric. Show by example that they may not be equal even for square matrices.

13. Show that for any wquare matrix $B, A = B + B^T$ is always symmetric and $K = B - B^T$ is aways skew-smmetric — which means that $K^T = -K$. Find these matrices when $B = \begin{bmatrix} 1 & 3 \\ 1 & 1\end{bmatrix}$, and write $B$ as the sum of a symmetric matrix and a skewsymetrix matrix.

1. How many entries can be chosen indeppendently, in a symmetric matrix of order $n$?
2. Hoe many entries can be chosen independently in a skew-symmetric matrix of order $n$?
1. If A = LDU, which 1’s on the diagonals of $L$ and $U$, what is the corresponding factorization of $A^T$? Note that $A$ and $A^T$ (squres matrices with no row exchanges share the same pivots.)
2. What triangula system will give solution to $A^Ty = b$
14. If $A=L_1D_1U_1$ and $A = L_2D_2U_2$, prove that $L_1 = L_2, D_1=D_2$ and $U_1 = U_2$, If $A$ is invertible the factorization is unique.

1. Derive the equation for $L_1^{-1}L_2D_2 = D_1U_1U_2^{-1}$ and explain why one side is lower triangular and the other side is upper triangular.
2. Compare the main diagonals in the eauation, and then compare the offdiagonals.
15. Under what conditions on its entries $A$ is invertible if: $A=\left[ \begin{array}{lll}{a} & {b} & {c} \\ {d} & {e} & {0} \\ {f} & {0} & {0}\end{array}\right] \quad \text { or } \quad A=\left[ \begin{array}{lll}{a} & {b} & {0} \\ {c} & {d} & {0} \\ {0} & {0} & {e}\end{array}\right] ?$

16. If the 3 by 3 matrix $A$ has row 1 + row 2 = row 3, show that it is impossible to solve $Ax = [ 1 \quad 2 \quad 4]^T$. Can $A$ be invertible?

17. Compute the symmetric $LDL^{T}$ factorization of $A=\left[ \begin{array}{ccc}{1} & {3} & {5} \\ {3} & {12} & {18} \\ {5} & {18} & {30}\end{array}\right] \quad \text { and } \quad A=\left[ \begin{array}{ll}{a} & {b} \\ {b} & {d}\end{array}\right]$

18. Find the inverse of $A=\left[ \begin{array}{cccc}{1} & {0} & {0} & {0} \\ \frac{1}4 & {1} & {0} & {0} \\ {\frac{1}{3}} & {\frac{1}{3}} & {1} & {0} \\ {\frac{1}{2}} & {\frac{1}{2}} & {1} & {1}\end{array}\right]$

19. If $A$ and $B$ are suqre matrices, show that $I - AB$ is invertible if $I - BA$ is invertible. Start from $B(I-AB) = (I-BA)B$.