Reduce the following matrix A to its normal form and hence find it’s rank, where

𝐴 = [

1 1 −1

1 −1 2

3 1 0

1

−1

1

]

To reduce the given matrix \( A \) to its normal form and determine its rank, we perform **Gaussian elimination** to row-reduce \( A \) to its row echelon form and then to its reduced row echelon form. Let's proceed.

### Step 1: Write down the matrix \( A \)
\[
A =
\begin{bmatrix}
1 & 1 & -1 \\
1 & -1 & 2 \\
3 & 1 & 0 \\
1 \\
-1 \\
1
\end{bmatrix}
\]

However, \( A \) appears inconsistent because it has two extra rows. Could you clarify if \( A \) is intended as a rectangular matrix (e.g., \( 4 \times 3 \))? Or confirm the structure so we can proceed correctly.

If the matrix \( A \) is supposed to be:
\[
A =
\begin{bmatrix}
1 & 1 & -1 \\
1 & -1 & 2 \\
3 & 1 & 0
\end{bmatrix}
\]
then we will process the \( 3 \times 3 \) matrix.

Let me know if this interpretation aligns with the matrix you meant!

Reduce the following matrix A to its normal form and hence find it’s rank, where

𝐴 = [
1 1 −1
1 −1 2
3 1 0
1
−1
1
]

Learn how to reduce a matrix to its normal form using Gaussian elimination and find its rank. This problem involves key concepts in linear algebra, including row echelon form and reduced row echelon form. Suitable for students in Grades 11-12.

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