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### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Find an equation involving $$g,\,h,$$and $$k$$ that makes this augmented matrix correspond to a consistent system:$$\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2}&5&{ - 9}&k\end{array}} \right]$$

The required equation is $$2g + h + k = 0$$.

See the step by step solution

## Step 1: Rewrite the augmented matrix

The given augmented matrix of a consistent system is as follows:

$$\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2}&5&{ - 9}&k\end{array}} \right]$$

## Step 2: Perform the elementary row operation

To eliminate the first term of the third row, perform an elementary row operation on the matrix $$\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2}&5&{ - 9}&k\end{array}} \right]$$, as shown below.

Add two times of row one to row three; i.e., $${R_3} \to {R_3} + 2{R_1}$$.

$$\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2 + 2\left( 1 \right)}&{5 + 2\left( { - 4} \right)}&{ - 9 + 2\left( 7 \right)}&{k + 2\left( g \right)}\end{array}} \right]$$

After the row operation, the matrix becomes:

$$\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\0&{ - 3}&5&{k + 2g}\end{array}} \right]$$

## Step 3: Apply the row operation

To eliminate the second term of the third row, add the second row to the third row; i.e., $${R_3} \to {R_3} - 3{R_1}$$.

$$\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\0&0&0&{k + 2g + h}\end{array}} \right]$$

## Step 4: Condition of a consistent system

For the system to be consistent, it should have unique or infinitely many solutions.

Let c denote the number $$k + 2g + h$$. Then the third equation, as represented by the augmented matrix above, is $$0 = b$$. This equation is possible if and only if b is equal to zero. It means the original system has a solution if and only if $$k + 2g + h = 0$$.

Hence, the required equation is $$2g + h + k = 0$$.