Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$\left[\begin{array}{llll} 1 & 3 & 0 & 0 \\ 0 & 0 & 1 & 8 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is in row-echelon form and in reduced row-echelon form.

Step 1: Verify Row-Echelon Form

Look at the given matrix. It can be seen that all zero rows (rows that contain only zeroes) are at the bottom of the matrix. Also, for each leading coefficient of a non-zero row (the first non-zero entry from the left), it is always to the right of the leading coefficient of the row above it. Therefore, it can be concluded that the given matrix is in row-echelon form.

Step 2: Check the Additional Requirements for Reduced Row-Echelon Form

For the given matrix to be in reduced row-echelon form, in addition to the requirements for row-echelon form, it also needs to satisfy the conditions that the leading coefficient in each row is 1 and is the only non-zero entry in its column. Looking at the matrix, the conditions are met: the leading coefficients are 1 and they are the only non-zero entries in their columns. This means that the given matrix is also in reduced row-echelon form.

Key Concepts

Matrix Transformations

Matrix transformations are fundamental operations used in linear algebra to simplify matrices and solve systems of equations. These transformations include row operations such as row swapping, scaling (multiplying a row by a non-zero constant), and row addition (adding or subtracting one row from another).

By applying matrix transformations, you can systematically convert any matrix into simpler forms like row-echelon form or even reduced row-echelon form. The aim is to make it easier to analyze the matrix or solve linear systems. Importantly, these operations must be performed with care to maintain the equivalence of the matrix to the original system represented.

• Row Swapping: Exchanging the position of two rows in a matrix.
• Scaling a Row: Multiplying all elements in a row by a non-zero scalar.
• Row Addition: Adding or subtracting the elements of one row from another.

Using these operations, one can rearrange a matrix's structure without changing its solutions. These transformations make matrices more 'friendly' to solve for variables.

Reduced Row-Echelon Form

The reduced row-echelon form (RREF) is a streamlined version of the row-echelon form of a matrix. It follows stricter rules to make solving systems even simpler. A matrix in RREF meets all the criteria of the row-echelon form and additional conditions.

To be in RREF, each leading coefficient (first non-zero number from the left in each non-zero row) must not only be 1, but it must also be the sole non-zero entry in its column. This ensures a clear path to solutions when working with systems of linear equations. Additionally, the leading coefficient of each row is positioned to the right of the leading coefficient in the previous row.

These criteria allow for easy identification of free variables and solutions in systems of equations, offering a form that directly relates to the solutions without further simplification needed. It's like having a perfectly formatted outline, where everything falls into place for immediate use.

Leading Coefficient

The leading coefficient in a matrix is crucial for understanding and transforming matrices into row-echelon or reduced row-echelon forms. It is the first non-zero number that appears from the left in a non-zero row. This concept helps determine the matrix's order and structure.

In the row-echelon form, each row's leading coefficient must be in a position to the right of the row above it. This creates a 'staircase' effect, making it clearer which variables are leading in the system of equations represented by the matrix.

In the reduced row-echelon form, the leading coefficients further need to be 1, making calculations straightforward. Moreover, they should be the only non-zero entries in their respective columns. This ensures that each row directly corresponds to a single variable, facilitating easy back-substitution in solving equations. Recognizing and manipulating leading coefficients is vital for transforming matrices and solving algebraic problems efficiently.