Find x if $\begin{bmatrix} x & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -2 & -3 \end{bmatrix} \begin{bmatrix} x \\ 3 \end{bmatrix} = 0$
Understand the Problem
The question asks you to find the value of 'x' that satisfies the given matrix equation. Essentially, you need to perform matrix multiplication and solve the resulting equation for 'x'.
Answer
$x = 1 \pm \sqrt{10}$
Answer for screen readers
$x = 1 + \sqrt{10}$ or $x = 1 - \sqrt{10}$
Steps to Solve
- Multiply the first two matrices
Given the expression $[x \quad 1] \begin{bmatrix} 1 & 0 \ -2 & -3 \end{bmatrix} \begin{bmatrix} x \ 3 \end{bmatrix} = 0$, first multiply the matrices $[x \quad 1]$ and $\begin{bmatrix} 1 & 0 \ -2 & -3 \end{bmatrix}$:
$[x \quad 1] \begin{bmatrix} 1 & 0 \ -2 & -3 \end{bmatrix} = [x(1) + 1(-2) \quad x(0) + 1(-3)] = [x-2 \quad -3]$
- Multiply the resulting matrix with the third matrix
Now multiply the result from step 1, $[x-2 \quad -3]$, by $\begin{bmatrix} x \ 3 \end{bmatrix}$:
$[x-2 \quad -3] \begin{bmatrix} x \ 3 \end{bmatrix} = (x-2)(x) + (-3)(3) = x^2 - 2x - 9$
- Solve the quadratic equation
We are given that the entire expression equals 0, so: $x^2 - 2x - 9 = 0$
Use the quadratic formula to solve for x: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -2$, and $c = -9$.
$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-9)}}{2(1)}$ $x = \frac{2 \pm \sqrt{4 + 36}}{2}$ $x = \frac{2 \pm \sqrt{40}}{2}$ $x = \frac{2 \pm 2\sqrt{10}}{2}$ $x = 1 \pm \sqrt{10}$
$x = 1 + \sqrt{10}$ or $x = 1 - \sqrt{10}$
More Information
The solutions to this matrix equation are $1 + \sqrt{10}$ and $1 - \sqrt{10}$, which are irrational numbers.
Tips
A common mistake is incorrectly performing matrix multiplication, especially with the signs. Another common mistake is incorrectly applying the quadratic forumla. Remember to carefully multiply each element and correctly identify a, b, and c for the quadratic formula.
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