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.

Answer: We start with the first system of equations

\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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