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Chapter 12: Problem 52

Solve for \(x\) \(\left|\begin{array}{ll}x & 1 \\ 3 & x\end{array}\right|=-2\)

Short Answer

Expert verified
The solutions are \(x = 1\) and \(x = -1\).

Step by step solution

01

- Understand the Determinant

To solve for \(x\) using the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), recall that the determinant is given by \(ad - bc\).
02

- Write the Determinant Formula

For the given matrix \(\begin{pmatrix} x & 1 \ 3 & x \end{pmatrix}\), identify \(a = x\), \(b = 1\), \(c = 3\) and \(d = x\).
03

- Substitute and Set the Determinant

Substitute into the determinant formula: \(ad - bc\). So we have: \(x \times x - 1 \times 3 = x^2 - 3\). This is equal to -2, according to the given equation: \(x^2 - 3 = -2\).
04

- Solve the Quadratic Equation

Isolate \(x^2\) by adding 3 to both sides: \(x^2 - 3 + 3 = -2 + 3\) which simplifies to \(x^2 = 1\).
05

- Find the Values of x

Take the square root of both sides to solve for \(x\): \(\begin{cases} x = 1 \ x = -1 \end{cases}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

2x2 matrix determinant
The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the form looks like this: \begin{pmatrix} a & b \ c & d \end{pmatrix}. To find the determinant, you use the formula: \(ad - bc\). Let鈥檚 break this down a bit more:
  • Identify the values of a, b, c, and d in the matrix. They are the elements of the matrix.
  • Multiply a by d and b by c.
  • Subtract bc from ad.
This will give you the determinant value. In our exercise, the matrix is \begin{pmatrix} x & 1 \ 3 & x \end{pmatrix}, which means a=x, b=1, c=3, and d=x. Substitute these values into the formula: \(x \times x - 1 \times 3 = x^2 - 3\). This helps us form an equation involving x.
quadratic equation
A quadratic equation is a type of polynomial equation of degree 2, generally written as \(ax^2 + bx + c = 0\). In the context of our exercise, solving the determinant lead us to the quadratic equation: \(x^2 - 3 = -2\).
  • We add 3 to both sides to isolate \(x^2\): \(x^2 - 3 + 3 = -2 + 3\).
  • This simplifies to \(x^2 = 1\).
Once the quadratic equation is simplified, we can solve for x by finding the square roots of both sides.
square root
Finding the square root is the process of determining what number, when multiplied by itself, gives the original number. For \(x^2 = 1\), we take the square root of both sides:
  • \(\begin{cases} x = 1 \ x = -1 \end{cases}\).
This means the solutions to the equation are \(x = 1\) and \(x = -1\). Remember that while every positive number has two square roots (one positive and one negative), the square root of zero is just zero.

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