Problem 17 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{C}=\left(\begin{array}{rrr} -5 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 3 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix is -105.

Step 1: Understand the Matrix Structure

The given matrix \( \mathbf{C} \) is a 3x3 diagonal matrix, which has non-diagonal elements equal to zero. Specifically, its entries are \( a_{11} = -5 \), \( a_{22} = 7 \), and \( a_{33} = 3 \), with all other elements being zero.

Step 2: Recall the Determinant of a Diagonal Matrix

For a diagonal matrix of the form \( \mathbf{C} = \begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix} \), the determinant is simply the product of the diagonal elements: \( \det(\mathbf{C}) = a \cdot b \cdot c \).

Step 3: Calculate the Product of Diagonal Elements

For matrix \( \mathbf{C} \), the diagonal elements are \(-5, 7,\) and \(3\). Calculate the product: \(-5 \times 7 \times 3 = -105\).

Step 4: Conclude the Determinant Value

The determinant of the matrix \( \mathbf{C} \) is \(-105\), based on the product of the diagonal elements.

Key Concepts

Diagonal Matrix

A diagonal matrix is a special kind of matrix where all the elements outside the main diagonal are zero. This makes it very easy to identify and work with. In a diagonal matrix, only the elements in the top-left to bottom-right diagonal can have values other than zero.
For example, consider

• The matrix
  \[\begin{bmatrix}2 & 0 & 0 \0 & 5 & 0 \0 & 0 & 9\end{bmatrix}\] is a diagonal matrix.

The values 2, 5, and 9 are on the main diagonal, and all other elements are zero. Diagonal matrices are useful in many mathematical operations because they simplify calculations considerably. When calculating a determinant or eigenvalues, having a diagonal matrix makes the process straightforward.
Simply focusing on the diagonal elements without worrying about all zeros in the off-diagonal positions reduces potential errors and complexities.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns. It is one of the simplest forms of square matrices beyond the 2x2 matrix, providing more complexity but still manageable for calculations.
Square matrices like the 3x3 matrix are important in linear algebra because they can be used to represent transformations and systems of equations. The general form of a 3x3 matrix is given by:

• \[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\]

This format makes it possible to describe a system of equations with three variables, which is often encountered in physics, computer graphics, and engineering problems.
When dealing with 3x3 matrices, operations like addition, scalar multiplication, and finding determinants or inverses become essential skills. Understanding how to manipulate these matrices is crucial for solving complex problems efficiently.

Determinant Calculation

Calculating the determinant of a matrix is an important step in many linear algebra problems. The determinant is a scalar value that is a property of square matrices. It provides information about the matrix, such as whether it is invertible.

• For a diagonal matrix, like the one in the exercise where \(\mathbf{C} = \begin{bmatrix}-5 & 0 & 0 \0 & 7 & 0 \0 & 0 & 3\end{bmatrix} \), the determinant is the product of the diagonal elements.

The computation simplifies to:

• \(-5 \times 7 \times 3 = -105\)

The determinant tells us that the matrix is invertible because it is non-zero.
Understanding determinant calculation can help students analyze properties of matrices, including solving linear systems, finding eigenvalues, and working on transformations in geometrical contexts. Knowing that diagonal matrices simplify this process helps in focusing on the essential elements quickly.