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

- Find the inverse of the permuation matrices $P=\left[ \begin{array}{lll}{0} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {0}\end{array}\right] \quad \text { and } \quad P=\left[ \begin{array}{lll}{0} & {0} & {1} \\ {1} & {0} & {0} \\ {0} & {1} & {0}\end{array}\right]$
- 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 $PP^T = I$.

From $AB=C$ find a formula $A^{-1}$. Do the same for $PA =LU$.

- If A is invertible and $AB =AC$ prove quickly that $B=C$.
- If $A = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$, find an example with $AB= AC$ but $B \neq C$.

If the inverse of $A^2$ is $B$ show that the inverse of $A$ is $AB$. (Thus whenever $A^2$ is invertible)

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

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

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

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?

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

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

- $A+B$ is not invertible although $A$ and $B$ are invertible
- $A + B$ is invertible although $A$ and $B$ are not invertible
- 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.

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 $\frac31$)

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

(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.

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.

- How many entries can be chosen indeppendently, in a symmetric matrix of order $n$?
- Hoe many entries can be chosen independently in a skew-symmetric matrix of order $n$?

- 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.)
- What triangula system will give solution to $A^Ty = b$

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.

- 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.
- Compare the main diagonals in the eauation, and then compare the offdiagonals.

Under what conditions on its entries $A$ is invertible if: $A=\left[ \begin{array}{lll}{a} & {b} & {c} \\ {d} & {e} & {0} \\ {f} & {0} & {0}\end{array}\right] \quad \text { or } \quad A=\left[ \begin{array}{lll}{a} & {b} & {0} \\ {c} & {d} & {0} \\ {0} & {0} & {e}\end{array}\right] ?$

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?

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

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

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