Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The values of \(x\) that satisfy the given equation are \(x = 1.5\) and \(x = -1.5\).

Step 1: Compute the determinant

To compute the determinant of a 2x2 matrix \[\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\] we subtract the product of the elements of the secondary diagonal from the product of the elements of the main diagonal. Therefore, for the given matrix \[\left[\begin{array}{cc} 2x & -3 \ -2 & 2x \end{array}\right]\] the determinant, denoted as \(det\), is calculated as:\[det = 2x * 2x - (-3)*(-2) = 4x^2 - 6\]

Step 2: Set the determinant equal to the given value

Next step is to set the calculated determinant equal to the given number 3 and solve the resulting equation. \[4x^2 - 6 = 3\]

Step 3: Solve for x

First rearrange the equation to treat it like a quadratic equation. \[4x^2 - 6 - 3 = 0\]\[4x^2 - 9 = 0\] Then divide by 4: \[x^2 - 2.25 = 0\] Eventually, solve for x by taking the square root of each side of the equation: \[√(x^2) = √±2.25\) Keep in mind that solving for x in this step produces two solutions: \[x = 1.5 , -1.5\]. Note that the ± symbol means that we have two solutions here. One positive and one negative.

Key Concepts

Determinants of Matrices

To understand matrices better, it's important to know about the determinant. In a 2x2 matrix or any matrix, the determinant is a special number that can tell us a lot about the matrix we're dealing with. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated by subtracting the product of the secondary diagonal from the product of the main diagonal. Specifically, for our example matrix: \[ \text{Determinant} = (2x \times 2x) - (-3 \times -2) \] This results in a determinant of \( 4x^2 - 6 \). The determinant gives us information about certain properties of the matrix, such as whether it is invertible or not. If a determinant is zero, the matrix isn't invertible, meaning a unique solution to the system of equations doesn’t exist.

Properties of Determinants

Determinants have some interesting properties that are very useful in solving mathematical problems. Here are a few key ones that are good to know:

• Sign Change Property: Swapping two rows or columns in a matrix changes the sign of its determinant. This means if you switch positions, your determinant value will flip positive to negative or vice versa.
• Multiplicative Property: The determinant of a product of two matrices is the product of their determinants. So, if you multiply matrices \( A \) and \( B \), then \( \det(AB) = \det(A) \cdot \det(B) \).
• Zero Determinant: If a matrix has a row or column of zeros, its determinant is zero, indicating it cannot be inverted.
• Determinant of Identity Matrix: The determinant of an identity matrix is always 1.

These properties are not just theoretical; they have practical applications in systems of linear equations, geometry, and even computer graphics. Knowing them can give you a deeper understanding of how matrices behave.

Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, which is any equation of the form \( ax^2 + bx + c = 0 \). It offers a way to find the solutions (also called roots) of the equation and is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's a quick breakdown:

• \( a \), \( b \), and \( c \): These are coefficients from your quadratic equation. The "\( x^2 \)" term is \( a \), "\( x \)" term is \( b \), and the constant is \( c \).
• Discriminant (\( b^2 - 4ac \)): This part of the formula determines the nature of the roots. If it's positive, you get two real and distinct solutions. If it is zero, there's exactly one real solution, and if negative, there are no real solutions, just complex ones.
• Plus and Minus (\( \pm \)): This symbol in the formula accounts for the two potential solutions you can get from a quadratic equation due to its parabolic nature.

While the quadratic formula is not needed for every quadratic equation in simple forms, it is a handy tool when factoring doesn’t work. It provides a surefire way to solve those more challenging equations.