Question

Solve the homogeneous linear system corresponding to the given coefficient matrix. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set x4 = t and x2 = s and solve for x1 and x3 in terms of t and s.)
[1 0 0 1]
[0 0 1 0]
[0 0 0 0]

(x1, x2, x3, x4) = _________

Answer 1

Answer:

$x_1 = -t$

$x_2 = s$

$x_3 = 0$

$x_4 = t$

Step-by-step explanation:

Given

$\left[\begin{array}{cccc}1&0&0&1\\0&0&1&0\\0&0&0&0\end{array}\right]$

Required

Determine x1 to x4

First, we write the augmented matrix

$\left[\begin{array}{cccc}1&0&0&1\\0&0&1&0\\0&0&0&0\end{array}\right] = \left[\begin{array}{c}0&0&0\end{array}\right]$

Taking the position of each column, we have:

$x_1 + 0 + 0 +x_4 = 0$

$0 + 0 + x_3 +0 = 0$

$0 + 0 + 0 + 0 = 0$

There is no explicit equation for $x_2$

So: $x_2 = s$ ----- arbitrary variable

$x_1 + 0 + 0 +x_4 = 0$ implies that:

$x_1 + x_4 = 0$

$0 + 0 + x_3 +0 = 0$ implies that

$x_3 = 0$

$0 + 0 + 0 + 0 = 0$ implies that

$0 = 0$

So, we have:

$x_1 + x_4 = 0$

$x_3 = 0$

$x_1 + x_4 = 0$ becomes

$x_1 = -x_4$

Let

$x_4 = t$

So:

$x_1 = -x_4$

$x_1 = -t$

Where t , s are real numbers