Question

The row-echelon form of the augmented matrix of a system of equations is given. Find the solution of the system.

Answer 1

x = -5; y = 4, z = 1

Explanation:

Given the row echelon form as:

`\left[\begin{array}{cccc}1&0&\ \ 4|&-1\\0&1&-1|&3\\0&0&\ \ 1|&1\end{array}\right]`

This matrix can be represented as:

`\left[\begin{array}{ccc}1&0&4\\0&1&-1\\0&0&1\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right] =\left[\begin{array}{c}-1\\3\\1\end{array}\right] \\\\Performing\ matrix\ multiplication\ gives:\\\\\left[\begin{array}{c}x+4z\\y-z\\z\end{array}\right] =\left[\begin{array}{c}-1\\3\\1\end{array}\right]`

Therefore:

z = 1

y - z = 3;

y = 3 + z = 3 + 1 = 4.

Hence, y = 4

x + 4z = - 1;

x = -1 - 4z = -1 - 4(1) = -5

x = -5