https://file.aiquickdraw.com/mathbot/file/1726155213208kfce1rhc.jpg

reduce the matrix into normal form and find its rank

To reduce the given matrix to its row echelon form (REF) and find its rank, let's first extract the matrix from the image.

The matrix is:

\[
\begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 \\
2 & 3 & 5 & 5 \\
3 & -4 & 5 & 8
\end{bmatrix}
\]

### Step 1: Perform row operations to reduce the matrix to row echelon form.
I'll walk through each operation.

#### Original matrix:

\[
\begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 \\
2 & 3 & 5 & 5 \\
3 & -4 & 5 & 8
\end{bmatrix}
\]

#### Row Operations:
1. Subtract row 1 from row 2:
   \[
   R_2 = R_2 - R_1 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 2 & 3 & 5 & 5 \\ 3 & -4 & 5 & 8 \end{bmatrix}
   \]

2. Subtract 2 times row 1 from row 3:
   \[
   R_3 = R_3 - 2R_1 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 3 & 3 \\ 3 & -4 & 5 & 8 \end{bmatrix}
   \]

3. Subtract 3 times row 1 from row 4:
   \[
   R_4 = R_4 - 3R_1 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 3 & 3 \\ 0 & -7 & 2 & 5 \end{bmatrix}
   \]

4. Subtract row 2 from row 3:
   \[
   R_3 = R_3 - R_2 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & -7 & 2 & 5 \end{bmatrix}
   \]

5. Add 7 times row 2 to row 4:
   \[
   R_4 = R_4 + 7R_2 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 16 & 26 \end{bmatrix}
   \]

6. Subtract 16 times row 3 from row 4:
   \[
   R_4 = R_4 - 16R_3 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 26 \end{bmatrix}
   \]

7. Divide row 4 by 26 to normalize:
   \[
   R_4 = \frac{1}{26} R_4 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}
   \]

### Row Echelon Form:
The matrix in row echelon form (REF) is:

\[
\begin{bmatrix}
1 & 1 & 1 & 1 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]

### Rank of the Matrix:
The rank of a matrix is the number of non-zero rows in its row echelon form. In this case, all 4 rows are non-zero. Hence, the rank of the matrix is **4**.

Would you like any further details or have any questions?

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Here are five questions related to this process:

1. What is the difference between row echelon form and reduced row echelon form?
2. How does the rank of a matrix affect the solutions of a system of linear equations?
3. Can a matrix have a rank greater than its number of rows or columns?
4. How do you find the determinant of a matrix, and how does rank relate to it?
5. What are some practical applications of matrix rank in data analysis?

**Tip**: To confirm the rank of a matrix, you can also use the determinant of the largest non-zero submatrix (if applicable).

Reduce the matrix into normal form and find its rank.

This tutorial provides a step-by-step guide to reducing a matrix into row echelon form and calculating its rank. Learn how to apply elementary row operations to solve matrix-related problems, useful for Grades 11-12 and undergraduate students.

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