Problem 26 - Precalculus

Question

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. $$\left[\begin{array}{lll|l} 1 & 0 & 1 & 3 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is in row-echelon form but not in reduced row-echelon form.

Step 1: Understand Row-Echelon Form

A matrix is in row-echelon form if: (1) Any rows of all zeroes are at the bottom. (2) The leading coefficient (also called the pivot) of a non-zero row is always to the right of the leading coefficient of the row above it. (3) All elements in the column below a leading coefficient are zero.

Step 2: Check for Row-Echelon Form

Based on the matrix:\[\begin{bmatrix}1 & 0 & 1 & 3 \0 & 1 & 2 & 2 \0 & 0 & 0 & 0\end{bmatrix}\]We observe that:- The bottom row is all zeroes.- The leading coefficient in the first row is in the first column, while the leading coefficient in the second row is in the second column.- All elements below the leading coefficients are zero.Therefore, the matrix is in row-echelon form.

Step 3: Understand Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if it is in row-echelon form and additionally: (1) The leading entry in each non-zero row is 1. (2) Each leading 1 is the only non-zero entry in its column.

Step 4: Check for Reduced Row-Echelon Form

For the matrix:\[\begin{bmatrix}1 & 0 & 1 & 3 \0 & 1 & 2 & 2 \0 & 0 & 0 & 0\end{bmatrix}\]- The leading coefficient in the first and second rows is 1, fulfilling the first condition.- However, the second condition is not fulfilled because the columns of the leading 1s (first and second columns) have other non-zero entries in the same column (e.g., 1 in the third column related to the leading 1 in the first row).Thus, the matrix is not in reduced row-echelon form.

Key Concepts

Matrix Transformations

Matrix transformations are operations applied to matrices that change their structure or form.
These transformations are crucial when solving systems of linear equations, among other applications. The primary operations in matrix transformations include:

• Swapping two rows.
• Multiplying a row by a nonzero constant.
• Adding or subtracting a multiple of one row to another row.

These operations enable us to convert a matrix into various forms, such as row-echelon form or reduced row-echelon form.
Each of these forms is pivotal for simplifying the matrix to make solving and interpreting the system of equations more straightforward.

Reduced Row-Echelon Form

The reduced row-echelon form (RREF) of a matrix is an advanced step beyond the row-echelon form.
In this form, each leading coefficient (or pivot) in a non-zero row is 1, and it is the only non-zero element in its column.

• Every leading entry of 1 has zeroes in all rows above and below it in its column.
• All rows consisting entirely of zeroes are at the bottom of the matrix.
• Each leading 1 appears to the right of the leading 1 in the previous row.

Reducing a matrix to its RREF simplifies the process of solving equations since each variable corresponds clearly to an entry in the matrix.
When you have your matrix in RREF, solutions to the system are more easily readable, sometimes directly giving you the values of variables.

Leading Coefficients

Leading coefficients, also known as pivots, are crucial elements in the manipulation of matrices.

• In a non-zero row, the first non-zero number is called the leading coefficient.
• These coefficients help to identify the structure of the matrix in both row-echelon form and reduced row-echelon form.
• They dictate the orientation of non-zero elements in parallel with each step of elementary row operations.

The position of leading coefficients helps in assessing if a matrix is in row-echelon form or reduced row-echelon form.
By understanding where these coefficients lie within your matrix, you can determine whether further simplification is needed and better interpret the solution tied to the system of equations.
Leading coefficients serve as markers for the correct ordering and position of variables when translating matrix rows back into algebraic equations.