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Introduction to Linear Algebra

Exercise 3.3.34

Answers

(a): If $s = (2, 3, 1, 0)$ is the only special solution to $Ax = 0$ , the complete solution is $x = cs$ (a line of solutions). The rank of $A$ must be $4 - 1 = 3 .$
(b): The fourth variable $x_{4}$ is not free in $s$ , and $R$ must be $[\begin{matrix} 1 & 0 & - 2 & 0 \\ 0 & 1 & - 3 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] .$
(c): $Ax = b$ can be solved for all $b$ , because $A$ and $R$ have full row rank $r = 3$ .

mathchakra

2022-01-23 15:15

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