Problem 31 - Algebra 2

Question

Solve det \(\left[\begin{array}{rrr}{4} & {x} & {-2} \\ {-x} & {-3} & {1} \\\ {-6} & {2} & {3}\end{array}\right]=-3\) for \(x\)

Step-by-Step Solution

Answer

No real solutions for \(x\).

Step 1: Understanding the Determinant

The determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is given by \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \). We can find the determinant of our given matrix by plugging in the corresponding elements.

Step 2: Setup the Determinant Formula

The matrix given is \( \begin{bmatrix} 4 & x & -2 \ -x & -3 & 1 \ -6 & 2 & 3 \end{bmatrix} \). Using the determinants formula, plug in the values: \( 4((-3)(3) - (1)(2)) - x((-x)(3) - (1)(-6)) - 2((-x)(2) - (-3)(-6)) \).

Step 3: Simplify Each Term

Calculate each term: \( 4((-9) - 2) = 4(-11) = -44 \), \( -x((3x) + 6) = -x(3x + 6) = -3x^2 - 6x \), and \( -2((-2x) - 18) = -2(-2x -18) = 4x + 36 \).

Step 4: Combine All Terms

Combine the simplified terms: \[-44 - 3x^2 - 6x + 4x + 36\]. This simplifies to \(- 3x^2 - 2x -8\).

Step 5: Set the Determinant to the Given Value

The problem states the determinant equals -3. Set the expression equal to -3: \[-3x^2 - 2x - 8 = -3\].

Step 6: Solve the Quadratic Equation

Rearrange the equation to standard quadratic form: \[-3x^2 - 2x - 8 + 3 = 0 \] which simplifies to \[-3x^2 - 2x - 5 = 0\].The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) can solve this, where \(a = -3\), \(b = -2\), and \(c = -5\).

Step 7: Compute the Quadratic Formula

Calculate the discriminant: \(b^2 - 4ac = (-2)^2 - 4(-3)(-5) = 4 - 60 = -56\).Since the discriminant is negative, there are no real solutions for \(x\).

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable, typically written as \(ax^2 + bx + c = 0\). In this expression, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\) since, if \(a = 0\), the equation would be linear, not quadratic. Quadratic equations are common in algebra and arise in various practical contexts, from physics to economics.To solve quadratic equations, we use different methods like factoring, completing the square, or the quadratic formula. However, in many cases, the quadratic formula is a widespread method due to its efficiency and straightforward application. This formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).In our exercise, simplifying the determinant of a 3x3 matrix results in the quadratic equation \(-3x^2 - 2x - 5 = 0\). Here, \(a = -3\), \(b = -2\), and \(c = -5\), which are substituted into the quadratic formula to seek solutions for \(x\).

Discriminant

The discriminant is a special part of solving quadratic equations, found within the quadratic formula. It is represented by the term \(b^2 - 4ac\). The discriminant helps to determine the nature of the roots of the quadratic equation, offering insight without solving it entirely first.The value of the discriminant reveals:

• If it is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, also known as a double root.
• If it is negative, no real solutions exist, because the square root of a negative number is not defined within the real number system.

For our matrix determinant equation, the discriminant is \((-2)^2 - 4(-3)(-5)\), which equals \(4 - 60 = -56\). Since this value is negative, the quadratic equation \(-3x^2 - 2x - 5 = 0\) has no real solutions.

Matrix Algebra

Matrix algebra involves operations on matrices, which are arrays of numbers organized in rows and columns similar to a spreadsheet. A 3x3 matrix, as utilized in our exercise, consists of 3 rows and 3 columns. Determinants are a specific measure from matrix algebra that gives important geometric properties including volume for matrices and solving linear systems of equations when combined with certain other techniques.The determinant of a 3x3 matrix \(A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\) is calculated using the formula:\[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]This formula might look complex, but it expands each element of the matrix’s first row across minors created from the remaining elements.In the given problem, finding and equating this determinant to \(-3\) leads to a quadratic equation, requiring further algebraic techniques to solve. Understanding matrix algebra, especially the determinant, is crucial for higher-level math and various applications in science and engineering.