## Linear Algebra and Its Applications, exercise 1.3.11

Exercise 1.3.11. Given the systems of equations $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&6 \\ u&+&2v&+&2w&=&11 \\ 2u&+&3v&-&4w&=&3 \end{array}$    and $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&7 \\ u&+&2v&+&2w&=&10 \\ 2u&+&3v&-&4w&=&3 \end{array}$

solve both systems using Gaussian elimination. $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&6 \\ u&+&2v&+&2w&=&11 \\ 2u&+&3v&-&4w&=&3 \end{array}$

The first elimination step produces $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&6 \\ &&v&+&w&=&5 \\ &&v&-&6w&=&-9 \end{array}$

The second elimination step produces $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&6 \\ &&v&+&w&=&5 \\ &&&&-7w&=&-14 \end{array}$

We then back-substitute, starting with solving for w: $\begin{array}{rcrcr}-7w = -14&\Rightarrow&w = 2&&\\v + w = 5&\Rightarrow&v + 2 = 5&\Rightarrow&v = 3\\u + v + w = 6&\Rightarrow&u + 3 + 2 = 6&\Rightarrow&u = 1\end{array}$

The solution to the first system of equations is thus u = 1, v = 3, w = 2.

We now go on to the second system of equations $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&7 \\ u&+&2v&+&2w&=&10 \\ 2u&+&3v&-&4w&=&3 \end{array}$

The first elimination step produces $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&7 \\ &&v&+&w&=&3 \\ &&v&-&6w&=&-11 \end{array}$

The second elimination step produces $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&7 \\ &&v&+&w&=&3 \\ &&&&-7w&=&-14 \end{array}$

We then back-substitute, starting with solving for w: $\begin{array}{rcrcr}-7w = -14&\Rightarrow&w = 2&&\\v + w = 3&\Rightarrow&v + 2 = 3&\Rightarrow&v = 1\\u + v + w = 7&\Rightarrow&u + 1 + 2 = 7&\Rightarrow&u = 4\end{array}$

The solution to the second system of equations is thus u = 4, v = 1, w = 2.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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