11760
294
21
​
  
0
1869
0
​
  
35280
−7840
63
​
  
∣
∣
∣
​
  
5997600
−348460
10710
​
  
con  esta matriz hacer estas operaciones 623f1−63f3→f1

623f2+14f3→f2

The matrix operations provided require elementary row operations, where we perform specific arithmetic transformations on the rows of the matrix.

### Given Matrix:
\[
\begin{pmatrix}
11760 & 294 & 21 \\
0 & 1869 & 0 \\
35280 & -7840 & 63 \\
5997600 & -348460 & 10710
\end{pmatrix}
\]

### Row Operations:
1. **\( 623f_1 - 63f_3 \to f_1 \)**: Multiply row 1 by 623, multiply row 3 by 63, subtract the result from row 1, and assign it back to row 1.
   
2. **\( 623f_2 + 14f_3 \to f_2 \)**: Multiply row 2 by 623, multiply row 3 by 14, add the result to row 2, and assign it back to row 2.

Let’s break down and calculate each operation step-by-step.

#### Step 1: Perform \( 623f_1 - 63f_3 \to f_1 \)
For row 1:
\[
f_1 = \begin{pmatrix} 11760 & 294 & 21 \end{pmatrix}
\]
Multiply by 623:
\[
623f_1 = \begin{pmatrix} 7314480 & 183162 & 13083 \end{pmatrix}
\]
For row 3:
\[
f_3 = \begin{pmatrix} 35280 & -7840 & 63 \end{pmatrix}
\]
Multiply by 63:
\[
63f_3 = \begin{pmatrix} 2222640 & -493920 & 3969 \end{pmatrix}
\]
Now subtract \( 63f_3 \) from \( 623f_1 \):
\[
f_1 = \begin{pmatrix} 7314480 - 2222640 & 183162 - (-493920) & 13083 - 3969 \end{pmatrix}
\]
\[
f_1 = \begin{pmatrix} 5091840 & 677082 & 9114 \end{pmatrix}
\]
Thus, the new row 1 is:
\[
f_1 = \begin{pmatrix} 5091840 & 677082 & 9114 \end{pmatrix}
\]

#### Step 2: Perform \( 623f_2 + 14f_3 \to f_2 \)
For row 2:
\[
f_2 = \begin{pmatrix} 0 & 1869 & 0 \end{pmatrix}
\]
Multiply by 623:
\[
623f_2 = \begin{pmatrix} 0 & 1164387 & 0 \end{pmatrix}
\]
For row 3:
\[
f_3 = \begin{pmatrix} 35280 & -7840 & 63 \end{pmatrix}
\]
Multiply by 14:
\[
14f_3 = \begin{pmatrix} 493920 & -109760 & 882 \end{pmatrix}
\]
Now add \( 14f_3 \) to \( 623f_2 \):
\[
f_2 = \begin{pmatrix} 0 + 493920 & 1164387 + (-109760) & 0 + 882 \end{pmatrix}
\]
\[
f_2 = \begin{pmatrix} 493920 & 1054627 & 882 \end{pmatrix}
\]
Thus, the new row 2 is:
\[
f_2 = \begin{pmatrix} 493920 & 1054627 & 882 \end{pmatrix}
\]

### Resulting Matrix:
After performing the operations, the updated matrix is:
\[
\begin{pmatrix}
5091840 & 677082 & 9114 \\
493920 & 1054627 & 882 \\
35280 & -7840 & 63 \\
5997600 & -348460 & 10710
\end{pmatrix}
\]

Would you like further details or have any questions?

Here are some related questions:
1. What are elementary row operations, and why are they useful in solving systems of equations?
2. Can this matrix be simplified further using row reduction?
3. How does Gaussian elimination utilize these operations?
4. How do determinants change after row operations?
5. Can this method be used to find the inverse of a matrix?

**Tip:** When performing row operations, always keep track of the order of operations to avoid mistakes!

con esta matriz hacer estas operaciones 623f1−63f3→f1 623f2+14f3→f2

Learn how to perform matrix row operations like 623f1 - 63f3 → f1 and 623f2 + 14f3 → f2 on a 4x3 matrix. This problem covers key linear algebra concepts, including row transformations and matrix manipulation. Suitable for university-level students studying linear algebra.

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