Problem 12 - Advanced Engineering Mathematics

Question

To determine whether the given matrix is singular or nonsingular. $$ \left(\begin{array}{lll} 0 & 2 & 0 \\ 0 & 0 & 1 \\ 8 & 0 & 0 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is nonsingular since its determinant is 16.

Step 1: Understand Matrix Determinants

A square matrix is singular if its determinant is zero. Otherwise, the matrix is nonsingular.

Step 2: Use the Formula for a 3x3 Matrix Determinant

For a 3x3 matrix \( \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \), the determinant is calculated as \( a(ei − fh) − b(di − fg) + c(dh − eg) \).

Step 3: Identify Matrix Elements

Identify the elements of the given matrix: \( a = 0 \), \( b = 2 \), \( c = 0 \), \( d = 0 \), \( e = 0 \), \( f = 1 \), \( g = 8 \), \( h = 0 \), \( i = 0 \).

Step 4: Calculate Each Term of the Determinant Formula

Calculate each part of the determinant formula:1. \( a(ei − fh) = 0(0 \times 0 - 1 \times 0) = 0 \).2. \( - b(di − fg) = -2(0 \times 0 - 8 \times 1) = 16 \).3. \( + c(dh − eg) = 0(0 \times 0 - 0 \times 8) = 0 \).

Step 5: Sum the Values to Calculate the Determinant

Add the values from the previous step: \( 0 + 16 + 0 = 16 \). The determinant is 16.

Key Concepts

Singular Matrix

A singular matrix is a square matrix with a determinant of zero. This means it does not have an inverse, making it a critical concept in linear algebra. Imagine a singular matrix as a flat object in the mathematical space that can't be "stretched" in any direction.
When a matrix has a determinant of zero, the rows or columns are linearly dependent. This dependency means one row is a combination of the others, conveying no new information.

• Key point: Singular matrices are important since their lack of invertibility impacts their applications in solving system equations.
• Remember: Having a determinant of zero is the hallmark of a singular matrix.

Nonsingular Matrix

In contrast, a nonsingular matrix has a non-zero determinant, meaning it has an inverse. If a matrix is nonsingular, its rows and columns are linearly independent. This independence is what allows nonsingular matrices to be used in solving a variety of mathematical problems.
A nonsingular matrix can represent rotations or transformations in space where dimensions are preserved and not flattened.

• Nonsingular matrices are key in calculations requiring matrix inverses, such as in process control, cryptography, and more.
• Essential detail: The matrix inverse is part of what makes nonsingular matrices so versatile.

Understanding whether a matrix is singular or nonsingular is vital for determining if it can be inverted.

3x3 Matrix Determinant Calculation

Calculating the determinant of a 3x3 matrix involves a specific formula that allows for an understanding of the matrix's properties. Let’s break it down so it's easy to grasp.
For a matrix \[\begin{pmatrix} a & b & c \d & e & f \g & h & i \end{pmatrix} \],
inserting the values into the determinant formula is key. The formula is:
\[a(ei − fh) − b(di − fg) + c(dh − eg)\]

• Start with identifying matrix elements. For instance, if your matrix has elements like a = 0, b = 2, etc., recognize each in their respective position.
• Calculate each term: Consider each multiplication and subtraction carefully to ensure accuracy. Each small error can lead to incorrect conclusions.
• Lastly, sum up the calculated parts to find the determinant, which reveals if the matrix is singular or nonsingular.

By carefully applying this formula, you can easily determine the characteristics of any 3x3 matrix.