The given matrix is:
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
The given matrix is:
$$
\begin{bmatrix}
1 & 0 & 3 & 0 \\
5 & 4 & 2 & 2 \\
2 & 5 & 1 & 0 \\
3 & 7 & 5 & 2
\end{bmatrix}
$$
Step 1: Make the entries below the leading 1 in the first column zero.
Perform the following row operations:
$R_2 \leftarrow R_2 - 5R_1$
$R_3 \leftarrow R_3 - 2R_1$
$R_4 \leftarrow R_4 - 3R_1$
The matrix becomes:
$$
\begin{bmatrix}
1 & 0 & 3 & 0 \\
0 & 4 & -13 & 2 \\
0 & 5 & -5 & 0 \\
0 & 7 & -4 & 2
\end{bmatrix}
$$
Step 2: Make the leading entry in the second row a 1.
Perform the row operation:
$R_2 \leftarrow \frac{1}{4}R_2$
The matrix becomes:
$$
\begin{bmatrix}
1 & 0 & 3 & 0 \\
0 & 1 & -\frac{13}{4} & \frac{1}{2} \\
0 & 5 & -5 & 0 \\
0 & 7 & -4 & 2
\end{bmatrix}
$$
Step 3: Make the entries below the leading 1 in the second column zero.
Perform the following row operations:
$R_3 \leftarrow R_3 - 5R_2$
$R_4 \leftarrow R_4 - 7R_2$
The matrix becomes:
$$
\begin{bmatrix}
1 & 0 & 3 & 0 \\
0 & 1 & -\frac{13}{4} & \frac{1}{2} \\
0 & 0 & \frac{45}{4} & -\frac{5}{2} \\
0 & 0 & \frac{75}{4} & -\frac{3}{2}
\end{bmatrix}
$$
Step 4: Make the leading entry in the third row a 1.
Perform the row operation:
$R_3 \leftarrow \frac{4}{45}R_3$
The matrix becomes:
$$
\begin{bmatrix}
1 & 0 & 3 & 0 \\
0 & 1 & -\frac{13}{4} & \frac{1}{2} \\
0 & 0 & 1 & -\frac{2}{9} \\
0 & 0 & \frac{75}{4} & -\frac{3}{2}
\end{bmatrix}
$$
Step 5: Make the entry below the leading 1 in the third column zero.
Perform the row operation:
$R_4 \leftarrow R_4 - \frac{75}{4}R_3$
The matrix becomes:
