Find Reduced Row Echelon Form [[2n],[n-1]]
In this problem, you are asked to transform a 2x1 matrix, where the first entry is 2n and the second entry is n-1, into its reduced row echelon form (RREF). The RREF of a matrix is a special form where each leading entry (first non-zero number from the left, also called a pivot) in a row is 1, and is the only non-zero entry in its column. Additionally, each leading entry is to the right of any leading entries in the rows above it, and any rows consisting entirely of zeros are at the bottom of the matrix. The task involves applying a series of elementary row operations to the matrix until it achieves these conditions. Elementary row operations include row swapping, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row.
$\left[\right. 2 n \\ n - 1 \left]\right.$
Answer
Solution:
Step 1:
Scale $R_1$ by multiplying with $\frac{1}{2n}$ to transform the first element into a $1$.
Step 1.1:
Apply the scalar multiplication to $R_1$: $\left[\frac{2n}{2n}, \frac{(n-1)}{2n}\right]^T$.
Step 1.2:
After simplification, $R_1$ becomes: $\left[1, \frac{n-1}{2n}\right]^T$.
Step 2:
To zero out the first element of $R_2$, execute $R_2 = R_2 - (n-1) \cdot R_1$.
Step 2.1:
Carry out the row operation on $R_2$: $\left[1, n-1 - (n-1) \cdot 1\right]^T$.
Step 2.2:
Post-simplification, $R_2$ is: $\left[1, 0\right]^T$.
Knowledge Notes:
The process described is for finding the Reduced Row Echelon Form (RREF) of a matrix. The RREF is a form of a matrix where each leading entry (the first non-zero number from the left in a row) is $1$, and is the only non-zero entry in its column. All rows consisting entirely of zeros are at the bottom of the matrix. The steps to achieve RREF include:
Row Scaling: Multiplying a row by a non-zero scalar. This is used to make the leading entry of a row $1$.
Row Addition/Subtraction: Adding or subtracting one row from another. This is used to create zeros in a column below or above a leading $1$.
Row Swapping: Interchanging two rows. This is not used in this particular problem but is often necessary in other cases to achieve RREF.
In LaTeX, matrices are often written using the $pmatrix$or $bmatrix$environments, and row operations are typically denoted by arrow symbols or by explicitly stating the operation, such as $R_2 = R_2 - (n-1)R_1$.
In this problem, the matrix is a $2 \times 1$ matrix, which simplifies the process since there is no need for row swapping or creating zeros above leading $1$s. The goal is to scale the first row to make the first element $1$ and then to subtract an appropriate multiple of the first row from the second row to make the second element $0$.
