Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rr} 1 & -3 \\ -1 & 8 \\ 0 & 4 \\ -2 & 10 \end{array}\right] $$

Step-by-Step Solution

Answer

The reduced row-echelon form of the provided matrix is the 4x2 matrix \[ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0\end{bmatrix}\].

Step 1: Identify the Pivot Element

The first pivot should be the number 1 in the top left of the matrix. The pivot lies at the intersection of the first row and the first column. \[\begin{{bmatrix}}1 & -3 \-1 & 8 \0 & 4 \-2 & 10\end{{bmatrix}}\]

Step 2: Zero Below the Pivot

By adding suitable multiples of the first row to the other rows, zeros can be placed below the pivot. R2 can be obtained by adding R1 to the original R2, R4 can be obtained by adding 2*R1 to the original R4. \[\begin{{bmatrix}}1 & -3 \0 & 5 \0 & 4 \0 & 4\end{{bmatrix}}\]

Step 3: Identify the Next Pivot Element

The next pivot element is number 5 in row 2. But in order to simplify the calculation the pivot element should be 1. The second row can be transformed, dividing it by 5. \[\begin{{bmatrix}}1 & -3 \0 & 1 \0 & 4 \0 & 4\end{{bmatrix}}\]

Step 4: Zero Above and Below the Second Pivot

Adding multiples of the new R2 to R1, R3 and R4 will result in zeros both above and below the second pivot. R3 can be obtained by subtracting 4*R2 from the original R3, R4 can be obtained by subtracting 4*R2 from the original R4, and R1 can be obtained by adding 3*R2 to the original R1. \[\begin{{bmatrix}}1 & 0 \0 & 1 \0 & 0 \0 & 0\end{{bmatrix}}\]

Step 5: Outcome in Reduced Row-Echelon Form

The matrix is now in Reduced Row-Echelon Form as there are leading 1s from left to right and there is zero below and above leading 1s.

Key Concepts

Matrix Operations

Matrix operations are fundamental processes in linear algebra, allowing us to manipulate matrices efficiently to solve various problems. Three primary operations are commonly used: addition, multiplication, and scalar multiplication. Matrix addition involves adding corresponding elements from two matrices of the same dimensions. Matrix multiplication requires a bit more understanding. It involves taking the dot product of the rows of the first matrix with the columns of the second. Only matrices where the number of columns in the first equals the number of rows in the second can be multiplied. When performing matrix operations, it’s crucial to follow the rules to achieve accurate results. Remember the order of operations, especially in multiplication, as it impacts the outcome. Scalar multiplication is simpler, as it involves multiplying every element in a matrix by a scalar (a single number). Understanding these operations is key, particularly when solving linear equations or transforming matrices into different forms, such as row-echelon or reduced row-echelon form.

Row-Echelon Form

Row-echelon form (REF) is a special way of arranging elements in a matrix. It simplifies the process of solving systems of linear equations. In REF, the matrix must meet specific criteria:

• All non-zero rows are above rows with all zero elements.
• Each leading coefficient (which is the first non-zero number from the left, in each row) is to the right of the leading coefficient in the row directly above it.
• All entries in a column below a leading coefficient are zero.

The process of getting a matrix into row-echelon form involves using row operations. These operations can include swapping rows, multiplying rows by a non-zero scalar, or adding/subtracting rows. Transforming a matrix into either row-echelon or reduced row-echelon form helps in identifying solutions to linear equations or in finding inverses of matrices. The journey to these forms helps ensure that you can easily isolate and solve variables systematically.

Pivot Element

The pivot element is a critical concept when dealing with matrices in linear algebra. In a matrix, the pivot is the first non-zero element in each row, and it is crucial during the transformation process into row-echelon or reduced row-echelon form. Why is a pivot element important?

• It is used as a reference point for operations to create zeros in the rest of the column.
• A pivot element of 1 makes calculations straightforward. For this reason, row operations might involve multiplying the row to create a leading 1.

The pivot element allows us to systematically clean up the matrix to make it easier to interpret or solve the system of equations it represents. During the process of row reduction, ensuring that each row has a pivot simplifies further calculations and provides a clear organizational structure to the transformed matrix. The ultimate goal in using pivot elements is to transform the matrix until it achieves either the row-echelon or reduced row-echelon form, facilitating the identification of solutions in a linear system.