Uploaded image

Q: Uploaded image

∻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. ⚹

Question

Uploaded image

Accepted Answer

∻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. ⚹

Sia