When Does This Matrix Have Zero, One and Infinite Solution?
$\Begingroup$ So I've Forgotten What the Conditions for When a Matrix Has Zero, One and Infinitely Many Solutions. Starting with This Matrix: $$ \Begin{Align}...
So I've forgotten what the conditions for when a matrix has zero, one and infinitely many solutions. Starting with this matrix: $$ \begin{align} &\left[\begin{array}{rrr|r} 1 & 1 & -1 & 0 \\ 2 & -1 & -5 & 3\\ -1 & 2 & a^2+3a & -3a \end{array}\right] \end{align} $$
I reduced this to: $$ \left[\begin{array}{rrr|r} 1 & 1 & -1 & 0 \\ 0 & -1 & -1 & 1\\ 0 & 0 & a^2+3a-4 & -3a+3 \end{array}\right] $$ But I'm not sure when this would have zero, one, or infinitely solutions.
Thanks in advance.
1 Answer
If $a^2 + 3a - 4 = 0$ and $-3a +3 =0$ then you will have infinitely many solutions.
If $a^2 + 3a - 4 = 0$ and $-3a+3\neq 0$, then you will not have any solutions. The point is that a row $(0\quad 0\quad 0\quad 1)$ will correspond to the equation $0x_1 + 0x_2 + 0x_3 = 1$.
If $a^2 + 3a - 4 \neq 0$ then you will have exactly one solution.
David Miller brings 15 years of experience in global economics, personal finance strategy, and market dynamics. He specializes in turning complex economic trends into actionable insights for everyday readers.
