Problem 18 - Advanced Engineering Mathematics

Question

In Problems 15-18, evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{D}=\left(\begin{array}{rrr} 0 & 7 & 0 \\ 4 & 0 & 0 \\ 0 & 0 & -2 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix \( \mathbf{D} \) is 0.

Step 1: Identify the Matrix Form

The given matrix \( \mathbf{D} \) is a 3x3 diagonal matrix. A diagonal matrix is one where all elements outside the main diagonal are zero, like \( \mathbf{D} \): \[ \mathbf{D} = \begin{pmatrix} 0 & 7 & 0 \ 4 & 0 & 0 \ 0 & 0 & -2 \end{pmatrix} \]

Step 2: Find the Determinant of the Diagonal Matrix

For a diagonal matrix, the determinant is the product of the diagonal elements. For \( \mathbf{D} \), the diagonal elements are 0, 0, and -2. Therefore, the determinant \( \text{det}(\mathbf{D}) \) is calculated as follows: \[ \text{det}(\mathbf{D}) = 0 \times 0 \times (-2) = 0 \]

Step 3: Confirm the Determinant is Second Property of Diagonal Matrices

The result is consistent with the property of determinants in diagonal matrices: if any diagonal entry is zero, the determinant of the matrix is zero. Hence, \( \text{det}(\mathbf{D}) = 0 \) is verified.

Key Concepts

Diagonal Matrix

A diagonal matrix is quite straightforward in structure. Imagine a square matrix where everything is zero, except for the values running diagonally from the upper left to the lower right. This is what we call a diagonal matrix. In math terms, a diagonal matrix has the form: \[ \begin{pmatrix} a_{11} & 0 & \cdots & 0 \ 0 & a_{22} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & a_{nn} \end{pmatrix}\] This means, apart from the main diagonal, every other entry is zero.
Why are diagonal matrices valued? They are much easier to work with for computations because of their simplicity. Whether you're adding, subtracting, or multiplying, diagonal matrices make the math cleaner. Plus, certain matrix operations like finding the determinant become more straightforward, as we'll explore.

• All non-diagonal elements are zero in a diagonal matrix.
• Diagonal elements are the focus for determinant calculations.

Matrix Properties

Understanding the various properties of matrices helps demystify many math operations. Matrices come with specific characteristics that guide their behavior in computations. When we talk about properties, we're often referring to aspects like:

• Determinant
• Identity matrices
• Orthogonality

For instance, a matrix is invertible if it has a non-zero determinant. But, as observed with diagonal matrices, if any diagonal element is zero, then the determinant goes to zero too.
Another crucial property is symmetry; some matrices have equal values mirrored across the diagonal, called symmetric matrices. However, the properties specifically related to diagonal matrices simplify over others by making several computations direct. For example, finding determinants or inverses becomes easier due to their structure.

Determinant Calculation

Calculating the determinant can sometimes seem complex, especially with larger matrices, but it isn't so with a diagonal matrix. In fact, for a diagonal matrix, the determinant is simply the product of its diagonal entries. This property drastically simplifies the math involved. Consider a 3x3 diagonal matrix:\[ \begin{pmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{pmatrix}\]The determinant of this matrix is calculated as \( a \times b \times c \). This rule emerges because the row operations lead directly to multiplying down the diagonal, ignoring everything else since all those entries are zero.

• Simplified calculation for diagonal matrices: \( product \ of \ diagonal \ elements\)
• If any diagonal element is zero, the entire determinant is zero.

This special case is handy, especially when dealing quickly with matrix determinants and predicting if the matrix is invertible or not.