Answer
See answers below
Work Step by Step
a) $\begin{bmatrix}
2&1 & 0 & 0\\
1& 2 & 0 & 0\\
0 & 0 & 3 & 4\\
0 & 0 & 4 & 3
\end{bmatrix} \approx^1
\begin{bmatrix}
1& 2 & 0 & 0\\
2&1 & 0 & 0\\
0 & 0 & 3 & 4\\
0 & 0 & 4 & 3
\end{bmatrix} \approx^2\begin{bmatrix}
1& 2 & 0 & 0\\
0&-3 & 0 & 0\\
0 & 0 & 3 & 4\\
0 & 0 & 1 & -1
\end{bmatrix} \approx^3 \begin{bmatrix}
1& 2 & 0 & 0\\
0&-3 & 0 & 0\\
0 & 0 & 1 & -1\\
0 & 0 & 3 & 4
\end{bmatrix} \approx^4 \begin{bmatrix}
1& 2 & 0 & 0\\
0&1 & 0 & 0\\
0 & 0 & 1 & -1\\
0 & 0 & 0 & 7
\end{bmatrix} \approx^5 \begin{bmatrix}
1& 2 & 0 & 0\\
0&1 & 0 & 0\\
0 & 0 & 1 & -1\\
0 & 0 & 0 & 1
\end{bmatrix} $
$1.P_{12}$
$2.A_{12}(-2),A_{34}(-1)$
$3.P_{34}$
$4.M_2(-\frac{1}{3}),A_{34}(-3)$
$5.M_4(\frac{1}{7})$
b) $rank(A)=42$
c) Use the GaussJordan Technique to determine the inverse of A:
$\begin{bmatrix}
2&1 & 0 & 0| 1 & 0 & 0 & 0\\
1& 2 & 0 & 0 | 0 & 1 & 0 & 0\\
0 & 0 & 3 & 4| 0 & 0 & 1 &0\\
0 & 0 & 4 & 3| 0&0 &0&1
\end{bmatrix} \approx^1
\begin{bmatrix}
1& 2 & 0 & 0|0 & 1 & 0 & 0\\
2&1 & 0 & 0|1 & 0 & 0 & 0\\
0 & 0 & 3 & 4|0 & 0 & 1 & 0\\
0 & 0 & 4 & 3 | 0 & 0 & 0 & 1
\end{bmatrix} \approx^2\begin{bmatrix}
1& 2 & 0 & 0 |0 & 1 & 0 & 0\\
0&-3 & 0 & 0 |1 & -2& 0 & 0\\
0 & 0 & 3 & 4|0 & 0 & 1 & 0\\
0 & 0 & 1 & -1|0 & 0 & -1 & 1
\end{bmatrix} \approx^3 \begin{bmatrix}
1& 2 & 0 & 0 |0 & 1 & 0 & 0\\
0&-3 & 0 & 0|1& -2 & 0 & 0\\
0 & 0 & 1 & -1|0 & 0 & -1 & 1\\
0 & 0 & 3 & 4|0 & 0 & 1 & 0
\end{bmatrix} \approx^4 \begin{bmatrix}
1& 2 & 0 & 0|0 & 1 & 0 & 0\\
0&1 & 0 & 0|-\frac{1}{3} & \frac{2}{3}& 0 & 0\\
0 & 0 & 1 & -1|0 & 0 & -1& 1\\
0 & 0 & 0 & 7|0 & 0 & 4 & -3
\end{bmatrix} \approx^5 \approx^6 \begin{bmatrix}
1& 2 & 0 & 0|\frac{2}{3} & \frac{-1}{3}& 0 & 0\\
0&1 & 0 & 0|-\frac{1}{3} & \frac{2}{3}& 0 & 0\\
0 & 0 & 1 &0|0&0& -\frac{3}{7} & \frac{4}{7}\\
0 & 0 & 0 & 1|0 &0& \frac{4}{7} & -\frac{3}{7}
\end{bmatrix} $
$1.P_{12}$
$2.A_{12}(-2),A_{34}(-1)$
$3.P_{34}$
$4.M_2(-\frac{1}{3}),A_{34}(-3)$
$5.M_4(\frac{1}{7}),A_{21}(-2)$
$6.A_{43}(1)$
Hence here, $A^{-1}=\begin{bmatrix}
\frac{2}{3} & \frac{-1}{3}& 0 & 0\\
-\frac{1}{3} & \frac{2}{3}& 0 & 0\\
0&0& -\frac{3}{7} & \frac{4}{7}\\
0 &0& \frac{4}{7} & -\frac{3}{7}
\end{bmatrix} $
