Find x if [x 1] * [[1, 0], [-2, -3]] * [[x], [3]] = 0

Steps to Solve

1. Matrix Multiplication 1: Multiplying the first two matrices

We first multiply the row matrix [x 1] with the 2x2 matrix: $\begin{bmatrix} 1 & 0 \ -2 & -3 \end{bmatrix}$. This gives us: $[x \ 1] \begin{bmatrix} 1 & 0 \ -2 & -3 \end{bmatrix} = [x(1) + 1(-2) \ \ x(0) + 1(-3)] = [x - 2 \ \ -3]$

2. Matrix Multiplication 2: Multiplying the result with the last matrix

Now, multiply the resulting row matrix $[x - 2 \ \ -3]$ with the column matrix $\begin{bmatrix} x \ 3 \end{bmatrix}$: $[x - 2 \ \ -3] \begin{bmatrix} x \ 3 \end{bmatrix} = (x - 2)(x) + (-3)(3) = x^2 - 2x - 9$

3. Solving the quadratic equation

We are given that the result of the entire multiplication is 0. Therefore, we have the quadratic equation: $x^2 - 2x - 9 = 0$

We can use the quadratic formula to solve for x: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -2$, and $c = -9$.

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-9)}}{2(1)} = \frac{2 \pm \sqrt{4 + 36}}{2} = \frac{2 \pm \sqrt{40}}{2} = \frac{2 \pm 2\sqrt{10}}{2} = 1 \pm \sqrt{10}$

Therefore, the two solutions for x are $1 + \sqrt{10}$ and $1 - \sqrt{10}$.