Problem 29 - Advanced Engineering Mathematics

Question

Find the values of \(\lambda\) that satisfy the given equation. $$ \left|\begin{array}{cc} -3-\lambda & 10 \\ 2 & 5-\lambda \end{array}\right|=0 $$

Step-by-Step Solution

Answer

\( \lambda = -4 \pm \sqrt{51} \).

Step 1: Understand the Determinant of a 2x2 Matrix

Recall that the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is given by the formula \( ad - bc \). In our matrix, the elements are \( a = -3-\lambda \), \( b = 10 \), \( c = 2 \), and \( d = 5-\lambda \).

Step 2: Set Up the Determinant Equation

Using the determinant formula, calculate the expression: \[ \text{det}(A) = (-3-\lambda)(5-\lambda) - (2)(10) \] Set this equal to zero because the determinant is given to be zero. \[ (-3-\lambda)(5-\lambda) - 20 = 0 \]

Step 3: Expand the Determinant Equation

First, expand the products: \[ (-3-\lambda)(5-\lambda) = (-3)(5) + (-3)(-\lambda) + (-\lambda)(5) + (-\lambda)(-\lambda) \]Simplify and combine like terms: \[ = -15 + 3\lambda + 5\lambda + \lambda^2 \]Thus, the equation becomes \[ \lambda^2 + 8\lambda - 15 - 20 = 0 \] or simplified: \[ \lambda^2 + 8\lambda - 35 = 0 \]

Step 4: Solve the Quadratic Equation

We need to solve the quadratic equation \( \lambda^2 + 8\lambda - 35 = 0 \). We can use the quadratic formula: \[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \( a = 1 \), \( b = 8 \), and \( c = -35 \).

Step 5: Calculate the Roots

Substitute the values into the quadratic formula: \[ \lambda = \frac{-8 \pm \sqrt{8^2 - 4(1)(-35)}}{2(1)} \]Calculate the discriminant: \[ 8^2 - 4(1)(-35) = 64 + 140 = 204 \]Then solve: \[ \lambda = \frac{-8 \pm \sqrt{204}}{2} \]Since \( \sqrt{204} = 2\sqrt{51} \), we simplify: \[ \lambda = \frac{-8 \pm 2\sqrt{51}}{2} \]\[ \lambda = -4 \pm \sqrt{51} \]

Step 6: Present the Values of λ

The values of \( \lambda \) that satisfy the equation are \( \lambda = -4 + \sqrt{51} \) and \( \lambda = -4 - \sqrt{51} \). These are the roots of the original determinant equation.

Key Concepts

2x2 Matrix

A 2x2 matrix is a square matrix that consists of two rows and two columns. In the context of this problem, it looks like this:

• The diagonal elements: upper left to bottom right (\(-3-\lambda\) and \(5-\lambda\))
• Off-diagonal elements: upper right to bottom left (\(10\) and \(2\))

Matrices are widely used in mathematics and engineering for various calculations, including finding systems of equations, transformations, and more.
The determinant is a unique number linked to square matrices that provides insightful properties. For a 2x2 matrix like this:\[\begin{pmatrix}a & b \c & d\end{pmatrix}\]The determinant is calculated by the formula \(ad - bc\). Understanding this formula is fundamental in finding eigenvalues and solving related equations.

Quadratic Equation

Quadratic equations are polynomial equations of the second degree. They typically take the form:\[ax^2 + bx + c = 0\]To solve a quadratic equation, you can use the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula helps find the values of \(x\) that satisfy the equation. In our instance with \(\lambda^2 + 8\lambda - 35 = 0\), the solving process entails finding two roots, which provide the numerical answers to the determinant problem.
Remember that the discriminant \(b^2 - 4ac\) plays a crucial role in determining the nature of the roots:

• If positive, there are two distinct real roots.
• If zero, there's exactly one real root.
• If negative, the roots are complex.

Eigenvalues

Eigenvalues are essential in various fields like physics and computer science, but particularly in linear algebra. They are scalars associated with a square matrix. In practical terms, they describe the amount a vector is stretched during a transformation represented by a matrix.
For a given 2x2 matrix:\[\begin{pmatrix}a & b \c & d\end{pmatrix}\]eigenvalues \(\lambda\) are obtained by solving the equation \(|A - \lambda I| = 0\), where \(I\) is the identity matrix. This translates to finding the determinant of \(A - \lambda I\), as was demonstrated in our problem.
Understanding eigenvalues helps with many applications in stability analysis, vibration analysis, finance models, and much more. They allow us to simplify complex systems and reveal insightful properties.