# How do I solve for $x$, when given three matrices multiplied together, equal to zero?

How do I solve for $x$, when given a $1 \times 3$, $3 \times 3$, and $3 \times 1$ matrix, all multiplied together set equal to zero with an $x$ variable in two of the matrices?

The question asks to determine values of $x$ such that:

$$\begin{bmatrix} x && 2 && 1 \end{bmatrix} \begin{bmatrix} 2 && 0 && -2 \\ 0 && 0 && 0 \\ -2 && -4 && -1 \end{bmatrix} \begin{bmatrix} x \\ -1 \\ 4 \end{bmatrix} = 0$$

Lets call the matrices $A$, $B$, and $C$ respectively as they are ordered. I started by multiplying matrix $A$ and $B$, and resulted in a $1 \times 3$ matrix $AB$:

$$\begin{bmatrix} 2x - 2 && -4 && -2x + 1 \end{bmatrix}$$

I then multiplied our new matrix with matrix $C$. The resultant $1 \times 1$ matrix was:

$$\begin{bmatrix} 2x^2 - 10x + 8 \end{bmatrix}$$

I then factored out a $2$ to get:

$$2 \begin{bmatrix} x^2 - 5x + 4 \end{bmatrix}$$

I then solved for values of $x$ that made it $0$, since the three matrices multiplied were initially set to equal $0$:

$$2 \begin{bmatrix} (x - 1)(x - 4) \end{bmatrix}$$

So it seems the problem is looking for the values $x = 1$ and $x = 4$, but upon submission it told me, the answers were wrong, so I feel I messed up somewhere or I’m missing other values. But I’ve gone through the problem a couple times and reached the same result. Can anyone point out my error or show me what I’m doing wrong?

$$[2x-2,-4,-2x-1]$$
$$ABC=(2x-2)x-1(-4)+4(-2x-1)=0 \implies 2x^2-10x=0$$