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Posted 26 days ago

Pattern Sequences

Pattern 1

The given matrix is:
$\begin{bmatrix}
0 & 0 & 1 & 2 \\
3 & -6 & -3 & 0 \\
8 & -12 & 2 & 11
\end{bmatrix}$

Pattern 2

To reduce the matrix, we can use row operations. The goal is to achieve row echelon form or reduced row echelon form

Pattern 3

Performing row operations, we can start by making the first element of the second row a leading 1. We can do this by dividing the second row by 3:
$R_2 \leftarrow \frac{1}{3} R_2$ resulting in:
$\begin{bmatrix}
0 & 0 & 1 & 2 \\
1 & -2 & -1 & 0 \\
8 & -12 & 2 & 11
\end{bmatrix}$

Pattern 4

Next, we can eliminate the first element of the third row by using the first row:
$R_3 \leftarrow R_3 - 8R_2$ resulting in:
$\begin{bmatrix}
0 & 0 & 1 & 2 \\
1 & -2 & -1 & 0 \\
0 & 4 & 10 & 11
\end{bmatrix}$

Pattern 5

Continuing with row operations, we can simplify the third row by dividing it by 4:
$R_3 \leftarrow \frac{1}{4} R_3$ resulting in:
$\begin{bmatrix}
0 & 0 & 1 & 2 \\
1 & -2 & -1 & 0 \\
0 & 1 & 2.5 & 2.75
\end{bmatrix}$

Pattern Rule

The pattern involves systematically applying row operations to reduce the matrix to a simpler form.

Answer

The reduced form of the matrix is achieved through a series of row operations, leading to a clearer representation of the relationships between the variables.

Key Concept

Matrix reduction using row operations

Explanation

The process of reducing a matrix involves applying row operations to simplify the matrix, making it easier to solve systems of equations or analyze the matrix's properties.

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