In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.

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Step 1: Start with the given augmented matrix: \[\left[\begin{array}{ccc|c} 1 & -1 & 1 & 8 \\ 2 & 3 & -1 & -2 \\ 3 & -2 & -9 & 9 \end{array}\right]\]

Step 2: Use the first row to eliminate the entries below the leading 1 in the first column. Specifically, replace row 2 with (row 2) - 2*(row 1), and row 3 with (row 3) - 3*(row 1). This will create zeros below the first pivot (1 in row 1, column 1).

Step 3: After these operations, the matrix becomes: \[\left[\begin{array}{ccc|c} 1 & -1 & 1 & 8 \\ 0 & 5 & -3 & -18 \\ 0 & 1 & -12 & -15 \end{array}\right]\] Here, the missing numbers in the second row, third column and augmented part are -3 and -18 respectively.

Step 4: Next, use the second row to eliminate the entry below the leading 1 in the second column. Replace row 3 with (row 3) - (1/5)*(row 2) to create a zero below the pivot in the second column.

Step 5: After this operation, the matrix becomes: \[\left[\begin{array}{ccc|c} 1 & -1 & 1 & 8 \\ 0 & 5 & -3 & -18 \\ 0 & 0 & -\frac{57}{5} & -\frac{57}{5} \end{array}\right]\] The missing numbers in the third row, third column and augmented part are \(-\frac{57}{5}\) and \(-\frac{57}{5}\) respectively.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Row-Echelon Form

Row-echelon form is a type of matrix form where all nonzero rows are above any rows of all zeros, the leading entry of each nonzero row is 1, and all entries below each leading 1 are zeros. This form simplifies solving systems of linear equations by back substitution.

Elementary Row Operations

Elementary row operations include swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting multiples of one row to another. These operations are used to transform a matrix into row-echelon form without changing the solution set of the corresponding system.

Augmented Matrix and System of Equations

An augmented matrix represents a system of linear equations, combining the coefficient matrix and constants into one matrix. Understanding how to manipulate this matrix helps in solving the system by applying row operations to isolate variables and find solutions.