Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 02 (b)

MATH152 April 2016

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Question B 02 (b)
In the system below, $x$, $y$, and $z$ are variables, and $a$ and $b$ are constants. ${\begin{array}{ccccccc}x&+&y&+&z&=&5\\x&~&~&+&z&=&1\\ax&~&~&+&z&=&b\end{array}}$ (b) Bring the augmented matrix to row echelon form.

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Hint
Recall the definition of row echelon form and find it by suitable Gaussian eliminations.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We operate Gaussian elimination on the rows and find row echelon form. First consider the second and the third row. To remove the entry in column 1 of row 3, we subtract off $-a$ times the second row; $\left({\begin{array}{ccc|c}1&1&1&5\\1&0&1&1\\a&0&1&b\end{array}}\right){\stackrel {\longrightarrow }{R_{3}-aR_{2}}}\left({\begin{array}{ccc|c}1&1&1&5\\1&0&1&1\\0&0&1-a&b-a\end{array}}\right).$ Then, we remove the entry in column 1 of row 2 by subtracting off $-1$ times the first row; $\left({\begin{array}{ccc|c}1&1&1&5\\1&0&1&1\\0&0&1-a&b-a\end{array}}\right){\stackrel {\longrightarrow }{R_{2}-R_{1}}}\left({\begin{array}{ccc|c}1&1&1&5\\0&-1&0&-4\\0&0&1-a&b-a\end{array}}\right).$ Finally, we obtain the row echelon form $\left({\begin{array}{ccc|c}1&1&1&5\\0&-1&0&-4\\0&0&1-a&b-a\end{array}}\right).$