Question

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.

Accepted Answer

Step 1: Start with the given augmented matrix:
$$\left[\begin{array}{ccc|c} 1 & -1 & 1 & 8 \ 2 & 3 & -1 & -2 \ 3 & -2 & -9 & 9 \end{array}ight]$$ 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:
$$\left[\begin{array}{ccc|c} 1 & -1 & 1 & 8 \ 0 & 5 & -3 & -18 \ 0 & 1 & -12 & -15 \end{array}ight]$$
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:
$$\left[\begin{array}{ccc|c} 1 & -1 & 1 & 8 \ 0 & 5 & -3 & -18 \ 0 & 0 & -\frac{57}{5} & -\frac{57}{5} \end{array}ight]$$
The missing numbers in the third row, third column and augmented part are $$-\frac{57}{5}$$ and $$-\frac{57}{5}$$ respectively.

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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Key Concepts

Row-Echelon Form

Elementary Row Operations

Augmented Matrix and System of Equations