数学代写|MA222 Linear algebra

(1) For each of the following matrices, use the Invertible Matrix Theorem to determine whether or not the matrix is invertible. Do not use any part of the theorem more than once (for instance, if you use part (b) to justify one of your answers, you may not use part (b) again). If the matrix is invertible, find its inverse.
(a) $\left[\begin{array}{ccc}4 & 2 & 1 \ 5 & 6 & 5 / 4 \ 8 & 5 & 2\end{array}\right]$
(b) $\left[\begin{array}{ccc}2 & 0 & 6 \ 6 & 1 & 6 \ -10 & -1 & -27\end{array}\right]$
(c) $\left[\begin{array}{cc}-4 & 6 \ 6 & -9\end{array}\right]$

\begin{prob}

\begin{prob}

True or false (no working needed, just circle the answer):
(a) T / F: If $A=\left[\begin{array}{ll}a & b \ c & d\end{array}\right]$ and $a d=b c$, then $A$ is not invertible.
(b) T $/ \mathrm{F}$ : If $A$ and $B$ are $n \times n$ invertible matrices, then the matrix equation $(A+B) \mathbf{x}=\mathbf{b}$ has a unique solution for each $\mathbf{b}$ in $\mathbb{R}^n$.
(c) T / F: the equation $A \mathbf{x}=\mathbf{b}$ has at least one solution for each $\mathbf{b}$ in $\mathbb{R}^n$ and if $A$ is invertible, then the solution is unique for each $b$.
(d) T / F: For $n \times n$ matrices $A$ and $B$, $\operatorname{det}(A+B)=\operatorname{det} A+\operatorname{det} B$.
(e) T / F: Each elementary matrix has determinant equal to \pm 1 .

Let $A=\left[\begin{array}{ccc}1 & 2 & -1 \ 1 & 2 & 2 / 3 \ 0 & 3 / 2 & -1 / 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}6 & -2 & 4 \ 0 & -5 & 3 \ 1 & -1 / 3 & 2 / 3\end{array}\right]$.
(a) Find $\operatorname{det} A$.
(b) Find $\operatorname{det} B$
(c) Which of the following are invertible? Justify your answer.
(i) $A B^T$
(ii) $(A B)^T$
(iii) $\left(A^T\right)^3$

(a) Solve the system.
(b) Write your solution as $\mathbf{x}=\mathbf{x}_p+\mathbf{x}_n$, where $\mathbf{x}_p$ is the particular solution of $A \mathbf{x}=\mathbf{b}$ and $\mathbf{x}_n$ is a linear combination of special solutions of $A \mathbf{x}=\mathbf{0}$.
(c) What is the rank of the coefficient matrix $A$ ?

$[10$ points $]$ Let $A=\left[\begin{array}{lll}1 & 1 & 0 \ 1 & 2 & 3 \ 4 & 5 & 3\end{array}\right]$ and $U=\left[\begin{array}{lll}1 & 1 & 0 \ 0 & 1 & 3 \ 0 & 0 & 0\end{array}\right]$. Which of the spaces $C(A), C(U), C\left(A^T\right)$, $C\left(U^T\right)$ are the same?

[10 points] Find the $L U$ factorization of $A$ and the complete solution to $A \mathbf{x}=\mathbf{b}$.
$$
A=\left[\begin{array}{rrrr}
1 & 1 & 2 & 0 \
0 & 1 & 1 & 1 \
-1 & 1 & 0 & 2
\end{array}\right], b=\left[\begin{array}{l}
1 \
2 \
3
\end{array}\right]
$$