Problem 7 - Advanced Engineering Mathematics

Question

State the appropriate theorem(s) in this section that justifies the given equality. Do not expand the determinants by cofactors. $$ \left|\begin{array}{rrrr} 0 & 5 & 0 & 6 \\ 2 & 1 & 0 & 8 \\ 0 & 2 & 0 & -9 \\ 0 & 6 & 0 & 4 \end{array}\right|=0 $$

Step-by-Step Solution

Answer

The determinant is zero because it has a zero column.

Step 1: Identify the Matrix Type

Examine the given matrix to identify any patterns or properties. Notice that the second column is the only non-zero column. All other columns contain at least one zero element.

Step 2: Recognize Applicable Theorem

Recall the theorem on determinants: If a matrix has a column (or row) that is a scalar multiple of another column (or row), the determinant of the matrix is zero.

Step 3: Apply the Theorem

In the given matrix, the third column is entirely zeros, making it unable to be a scalar multiple of any other column. Instead, recognize that if any entire row or column of a square matrix is zero, the determinant is zero. In this matrix, the third column is zero.

Key Concepts

Determinant Properties

The determinant of a matrix is a special number that can tell us a lot about the matrix itself. There are several properties of determinants that can make computing or understanding them easier:

• Row and column swap: Swapping two rows or two columns of a matrix flips the sign of the determinant.
• Proportional rows or columns: If two rows or columns of a matrix are proportional (one is a scalar multiple of the other), the determinant is zero.
• Additive property: The determinant is linear across any row or column. This means you can separate additions inside the determinants into two separate determinant calculations.
• Zero row or column: If an entire row or entire column is zeros, the determinant is zero.
• Identity properties: The determinant of an identity matrix is 1, which provides a baseline for other determinant calculations.

Understanding these properties can greatly help in solving determinant-related problems effectively. They provide quick insights and shortcuts to check and calculate determinants without expanding via cofactors.

Zero Column Determinant

The determinant of a matrix becomes particularly easy to calculate if one of its columns or rows is entirely zero. This is because it follows a simple property:

• If any column or row of a matrix is completely filled with zeros, the determinant of that matrix is zero.

This property makes solving determinant problems faster, as we do not have to perform cofactor expansions, which can be lengthier and more error-prone. In practice, as soon as we spot a zero column or row, we can confidently claim that the determinant is zero without further computations. Recognizing this property early in the process can save time and simplify work while studying or solving determinant problems.

Matrix Theory

Matrix theory forms the basis for many calculations and applications in linear algebra. A matrix is essentially a rectangular grid of numbers and can be used to represent data or systems of linear equations. Knowing about matrices and their determinants is crucial because:

• Matrices help solve linear equations by representing them in a compact form. This is particularly useful in computational software and for algorithmic solutions.
• The determinant provides insights into some properties of a matrix, such as whether it is invertible. A matrix with a non-zero determinant can generally be inverted, which is essential for solving equations.
• Determinants help us understand geometric properties, like the volume scaling of transformations described by matrices.

Matrix theory is foundational for more advanced topics in mathematics, physics, engineering, and computer science. By grasping the basics of matrix theory and properties like determinants, students can unlock a deeper understanding of these fields and appreciate the structure and solutions that matrix theory provides.