Problem 11 - Advanced Engineering Mathematics

Question

In Problems 9-14, evaluate the determinant of the given matrix. $$ \left(\begin{array}{rr} 3 & 5 \\ -1 & 4 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix is 17.

Step 1: Identify the Matrix

The given matrix is a 2x2 matrix: \( A = \begin{pmatrix} 3 & 5 \ -1 & 4 \end{pmatrix} \). We are asked to evaluate its determinant.

Step 2: Apply Determinant Formula for 2x2 Matrix

For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as: \( \text{det}(A) = ad - bc \).

Step 3: Substitute the Values

Substitute the corresponding values into the determinant formula. Here, \( a = 3 \), \( b = 5 \), \( c = -1 \), \( d = 4 \). The formula becomes: \( \text{det}(A) = 3 \times 4 - 5 \times (-1) \).

Step 4: Calculate the Products

Calculate the products in the formula: \( 3 \times 4 = 12 \) and \( 5 \times (-1) = -5 \).

Step 5: Simplify the Expression

Substitute the products back into the expression: \( \text{det}(A) = 12 - (-5) \). Simplify to get \( \text{det}(A) = 12 + 5 \).

Step 6: Find the Determinant

Add the values to find the determinant: \( \text{det}(A) = 17 \).

Key Concepts

Matrix Operations

Matrices are a fundamental concept in linear algebra and enable numerous useful computations, such as solving systems of equations, transformations, and more. When we perform matrix operations, we are manipulating these arrays of numbers according to certain rules to produce new matrices or single values, known as scalars. Some key matrix operations include:

• Addition and Subtraction: Only possible between matrices of the same dimensions, performed element-wise.
• Multiplication: Can be more complex; for two matrices to be multiply-able, the number of columns in the first must equal the number of rows in the second.
• Determinant Calculation: Provides a scalar value representing certain properties of the matrix, such as its invertibility.

Matrix operations, including finding determinants, are essential to exploit the power of matrices in various applications in science and engineering.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices, consisting of two rows and two columns.It's generally represented as:\[\begin{pmatrix}a & b \c & d\end{pmatrix}\]where \(a\), \(b\), \(c\), and \(d\) are elements of the matrix. These matrices form the basic building blocks for larger matrices and are often used as introductory examples to illustrate matrix concepts due to their simplicity. The 2x2 matrix is a perfect entry point to explore matrix operations, as calculating their determinant is straightforward and helps to build understanding about how more complex matrices work.

Determinant Calculation

The determinant of a matrix is a special scalar value that can indicate certain properties of the matrix. For example, in a 2x2 matrix such as \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant helps in determining:

• Whether the matrix is invertible—non-zero determinant means it is.
• Volume scaling factor of the linear transformation represented by the matrix.

The formula to find the determinant of a 2x2 matrix is simple: \( \text{det}(A) = ad - bc \). This involves multiplying the diagonal elements, \(a\) and \(d\), and subtracting the product of the off-diagonal elements, \(b\) and \(c\). In our example, \[ \text{det}(A) = 3 \times 4 - 5 \times (-1) = 12 + 5 = 17 \]. This determinant value tells us that the transformation represented by this matrix scales volumes by a factor of 17 and signifies invertibility.