Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{rrr|r}1 & 0 & 1 & 3 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & -1 & 6\end{array}\right]\)

Step-by-Step Solution

Answer

The given matrix is \(\left[\begin{array}{rrr|r}1 & 0 & 1 & 3 \\\ 0 & 1 & 0 & 4 \\\ 0 & 0 & -1 & 6\end{array}\right]\). Although most conditions for a matrix to be in row-reduced form are met, the third pivot in this matrix is -1, which does not meet the condition that the leftmost nonzero entry of a non-zero row should be 1. Therefore, the matrix is not in row-reduced form.

Step 1: Determine if all rows containing all zeros are at the bottom of the matrix.

There are no rows with all zeros in the given matrix. Thus, this condition is met.

Step 2: Determine if the leftmost nonzero entry of a non-zero row (pivot) is 1.

In the given matrix, the pivots are the first nonzero entries of each non-zero row, which are 1, 1, and -1. The first two pivots are 1, but the third pivot is -1, which does not meet this condition. Thus, the matrix is not in row-reduced form.

Step 3: Determine if the pivot of each nonzero row is located to the right of the pivot of the row above it.

In the given matrix, the pivots are located as follows: - Row 1: Pivot at position (1,1) - Row 2: Pivot at position (2,2) - Row 3: Pivot at position (3,3) The pivot of row 2 is to the right of the pivot of row 1 and the pivot of row 3 is to the right of the pivot of row 2. Thus, this condition is met.

Step 4: Determine if all entries in the columns above and below the pivots are zeros.

In the given matrix, the entries above and below the pivots are already zeros. Thus, this condition is met. Since the second condition is not met, the matrix is not in row-reduced form.

Key Concepts

Pivot Positions in Matrices

Understanding pivot positions in matrices is crucial for grasping the concepts behind various matrix operations. A pivot, or pivotal entry, is essentially the first non-zero number in a row, when reading from left to right. Pivot positions are integral to matrix transformations, and they play a significant role in finding solutions to systems of linear equations.

When converting a matrix to row-reduced echelon form (RREF), each pivot should ideally be '1', and it is called a leading '1'. Moreover, pivot positions must fulfill additional requirements to achieve row-reduced echelon form. These include having all non-pivotal elements in the pivot's column be zero, which ensures that pivotal variables represent their unique equation in the system.

In the provided exercise, the third row does not meet the pivotal criterion because its leading entry is '-1' instead of '1'. This highlights that the matrix is not fully in row-reduced form. Correctly identifying and managing pivot positions is essential when using methodical approaches, like Gaussian elimination, to solve systems of equations.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations, and it's one of the most widely taught techniques in algebra. It involves performing a series of operations to transform a matrix into its row-reduced echelon form. This method simplifies the system, making it easier to solve for the unknowns.

The process begins by forming an augmented matrix that includes the coefficients of the variables as well as the constants from the equations. The next steps include:

• Eliminating lower entries below the pivots to form zeros by using elementary row operations.
• Ensuring pivots are all '1's and creating zeros above these pivots to isolate each variable.
• Continuing the process for each row until the matrix is in row-reduced echelon form.

Keep in mind that while Gaussian elimination streamlines the solving process, the final solution is only as accurate as the execution of each step. In our textbook solution, the matrix did not achieve RREF due to the third row's pivot being '-1' instead of '1'. To reach RREF, additional steps would be required to adjust this pivot.

Elementary Row Operations

Elementary row operations are the tools we use to manipulate matrices into row-reduced echelon form, and they include three specific types of operations:

• Row switching - swapping two rows.
• Row multiplication - multiplying all elements of a row by a non-zero scalar.
• Row addition - adding or subtracting the elements of one row from another row.

These transformations must maintain the system's consistency, which means they do not change the solutions to the system of equations. These operations are used iteratively throughout Gaussian elimination and are crucial in achieving the row-reduced echelon form of a matrix.

The proper use of elementary row operations is evident in the step-by-step solution where the pivots are isolated through these operations. However, the process was incomplete since the third row's pivot was not converted to '1'. This indicates the necessity to further apply row multiplication to turn that '-1' into a '1', thus moving the matrix closer to RREF.