Problem 14 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix. $$ \left(\begin{array}{cc} -3-\lambda & -4 \\ -2 & 5-\lambda \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant is \( \lambda^2 - 2\lambda - 23 \).

Step 1: Identify the Formula for 2x2 Determinant

For a 2x2 matrix of the form \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as \( ad - bc \). In this problem, \( a = -3-\lambda \), \( b = -4 \), \( c = -2 \), and \( d = 5-\lambda \).

Step 2: Apply the Determinant Formula

Substitute the values into the determinant formula: \(-3-\lambda \) for \( a \), \( -4 \) for \( b \), \( -2 \) for \( c \), and \( 5-\lambda \) for \( d \). The determinant is:\[det(A) = (-3-\lambda)(5-\lambda) - (-4)(-2)\]

Step 3: Simplify the Expression

First, expand the product \((-3-\lambda)(5-\lambda)\):\[(-3-\lambda)(5-\lambda) = (-3) \cdot (5) + (-3)(-\lambda) + (-\lambda)(5) + (-\lambda)(-\lambda) \]\[= -15 + 3\lambda - 5\lambda + \lambda^2\]\[= \lambda^2 - 2\lambda - 15\]Then calculate the product \((-4)(-2)\):\[ (-4)(-2) = 8\]

Step 4: Calculate the Determinant

Substitute the simplified expression back into the determinant formula:\[det(A) = \lambda^2 - 2\lambda - 15 - 8 = \lambda^2 - 2\lambda - 23\]

Step 5: Present the Final Answer

The determinant of the matrix \( \begin{pmatrix} -3-\lambda & -4 \ -2 & 5-\lambda \end{pmatrix} \) is \( \lambda^2 - 2\lambda - 23 \).

Key Concepts

2x2 Matrices

In mathematics, a 2x2 matrix is a simple square matrix with two rows and two columns. Each element in the matrix can be written as a specific entry, typically denoted as \ a, b, c, \ and d. Understanding the structure of a 2x2 matrix is crucial for matrix operations like addition, multiplication, and especially finding determinants.
2x2 matrices are particularly important because they represent basic linear transformations in a plane.
When dealing with 2x2 matrices, you essentially deal with linear equations or transformations that can be visualized in two-dimensional space.

• The first row typically consists of elements \(a\) and \(b\).
• The second row consists of elements \(c\) and \(d\).

These matrices can also represent simple systems of linear equations or transformations that can involve scaling, rotating, or shearing in a two-dimensional plane.

Determinant Formula

The determinant of a 2x2 matrix is a special number calculated using a specific formula. This formula helps determine certain properties of the matrix, such as invertibility and the area scaling factor for linear transformations.
For a matrix\[\begin{pmatrix}a & b\ c & d\end{pmatrix}\],
the determinant is calculated using:
\[ det(A) = ad - bc \]

• This formula multiplies the 'a' and 'd' components diagonally and subtracts from it the product of 'b' and 'c'.
• The result helps in understanding whether a matrix can be inverted.

In our exercise, the terms are represented as \(a = -3-\lambda\), \(b = -4\), \(c = -2\), and \(d = 5-\lambda\). By plugging these values into the determinant formula, we can solve it step by step to get a polynomial expression in terms of \(\lambda\). This polynomial shows how changes in the variable \(\lambda\) affect the determinant's value.

Eigenvalues

The concept of eigenvalues is essential when dealing with matrices. Eigenvalues are closely related to the determinant and are used to understand the properties of matrix transformations.
Eigenvalues are specific scalars \(\lambda\) that satisfy the equation:\[ det(A - \lambda I) = 0 \]

• The equation involves the determinant of the matrix when it is subtracted by \(\lambda\) times the identity matrix \(I\).
• The given matrix's determinant expressed as \(\lambda^2 - 2\lambda - 23\) helps find eigenvalues by setting entire expression to zero \(\lambda^2 - 2\lambda - 23 = 0\) and solving for \(\lambda\).

Finding these values help in understanding different modes or directions in which the transformation represented by the matrix acts.
If the determinant equals zero, no inverse exists, and the matrix is said not to be full rank.