\[\begin{matrix}
\end{matrix}
\textbf{Problem set 1}\\
\text{1. What is the rank of the matrix A?}\\
A =\begin{bmatrix}
1 & 2 & 3 & 4 \\
-1 & 0 & 1 & 3 \\
0 & 1 & -2 & 1 \\
5 & 4 & -2 & -3
\end{bmatrix}\\
\begin{array}{c}
R_2 - (-1) \times R_1 \rightarrow R_2
\end{array}
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & 2 & 4 & 7 \\
0 & 1 & -2 & 1 \\
5 & 4 & -2 & -3
\end{bmatrix}
\begin{array}{c}
R_4 - 5 \times R_1 \rightarrow R_4
\end{array}
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & 2 & 4 & 7 \\
0 & 1 & -2 & 1 \\
0 & -6 & -17 & -23
\end{bmatrix}\\
\begin{array}{c}
R_3 - \frac{1}{2} \times R_2 \rightarrow R_3
\end{array}
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & 2 & 4 & 7 \\
0 & 0 & -4 & \frac{-5}{2} \\
0 & -6 & -17 & -23
\end{bmatrix}
\begin{array}{c}
R_4 - (-3) \times R_2 \rightarrow R_4
\end{array}
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & 2 & 4 & 7 \\
0 & 0 & -4 & \frac{-5}{2} \\
0 & 0 & -5 & -2
\end{bmatrix}\\
\begin{array}{c}
R_4 - \frac{5}{4} \times R_1 \rightarrow R_4
\end{array}
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & 2 & 4 & 7 \\
0 & 0 & -4 & \frac{-5}{2} \\
0 & 0 & 0 & \frac{9}{8}
\end{bmatrix}
\text{The rank of Matrix $A$ = 4}\\
\text{2.) Given an mxn matrix where m > n, what can be the maximum
rank?} \\ \text{The mini- mum rank, assuming that the matrix is
non-zero?} \\
\text{The maximum rank of an \( m \times n \) matrix where \( m > n
\) is \( n \). }\\
\text{The minimum rank of a non-zero matrix is 1}\\
\\[1em]
\textbf{3. What is the rank of the matrix B?}\\
\begin{align*}
\begin{bmatrix}
1 & 2 & 1 \\
3 & 6 & 3 \\
2 & 4 & 2
\end{bmatrix}
\begin{array}{c}
R_2 - 3 \times R_1 \rightarrow R_2
\end{array}
\begin{bmatrix}
1 & 2 & 1 \\
0 & 0 & 0 \\
2 & 4 & 2
\end{bmatrix}
\begin{array}{c}
R_3 - 2 \times R_1 \rightarrow R_3
\end{array}
\begin{bmatrix}
1 & 2 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\end{align*}
\\[1em]
\textbf{Problem set 2}\\
\begin{bmatrix}
1-\lambda & 2 & 2 \\
0 & 4-\lambda & 5 \\
0 & 0 & 6-\lambda
\end{bmatrix}\\
det(A-λI)
=\begin{bmatrix}
-\lambda+1 & 2 & 2 \\
0 & -\lambda+4 & 5 \\
0 & 0 & -\lambda+6
\end{bmatrix} =
(-λ+1)(-λ+4)(-λ+6)
=-λ^3+11λ^2-34λ+24\\
\begin{array}{c}
\lambda_1 = 1, \lambda_2 = 4, \lambda_3 = 6
\end{array}\\
v_1= \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \lambda_1 = 1\\
v_2 = \begin{bmatrix}
\frac{2}{3} \\
1 \\
0
\end{bmatrix} , \lambda_2 = 4\\
v_3 =\begin{bmatrix}
\frac{7}{5} \\
\frac{5}{2} \\
1
\end{bmatrix}, \lambda_1 = 6\]