Problem 8 - Advanced Engineering Mathematics

Question

To determine whether the given matrix is singular or nonsingular. $$ \left(\begin{array}{rrr} 2 & 3 & 0 \\ 0 & 11 & 14 \\ -1 & 4 & 7 \end{array}\right) $$

Step-by-Step Solution

Answer

The given matrix is singular.

Step 1: Understand the Problem

A matrix is considered singular if its determinant is zero. Conversely, it is nonsingular if its determinant is not zero. To solve this problem, we need to find the determinant of the given matrix.

Step 2: Matrix Formulation

We are given the matrix:\[\begin{pmatrix}2 & 3 & 0 \0 & 11 & 14 \-1 & 4 & 7 \end{pmatrix}\]We need to find its determinant to determine if it is singular or nonsingular.

Step 3: Calculate the Determinant

For a 3x3 matrix \( \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \), the determinant is calculated as:\[ Determinant = a(ei - fh) - b(di - fg) + c(dh - eg) \]Applying this formula to our matrix, we have:\[= 2(11 \times 7 - 14 \times 4) - 3(0 \times 7 - 14 \times (-1)) + 0(0 \times 4 - 11 \times (-1))\]

Step 4: Evaluate the Expression

Continue to simplify the determinant expression:For the first term: \[2(77 - 56) = 2 \times 21 = 42 \]For the second term:\[-3(0 + 14) = -42 \]For the third term, since it’s multiplied by zero, it is \(0\).Now the total determinant is:\[42 - 42 + 0 = 0\]

Step 5: Conclude the Type of Matrix

Since the determinant equals zero, the matrix is singular. A singular matrix has no inverse, and therefore, it's not full rank.

Key Concepts

Singular Matrix

A singular matrix is a matrix that does not have an inverse. One of the key signs that a matrix is singular is that its determinant is zero.
When dealing with matrices, knowing whether a matrix is singular is crucial because a singular matrix cannot be used in certain matrix operations, like finding a solution to a system of linear equations.

• Zero Determinant: The main characteristic of a singular matrix is that its determinant equals zero.
• Non-Invertible: Since it has no inverse, you cannot use it in calculations where inverses are needed, such as solving linear systems using matrix methods.

In practical terms, a singular matrix could indicate that the system represented by the matrix is dependent and doesn't have a unique solution. Understanding whether a matrix is singular helps in determining the set of operations you can safely perform.

Nonsingular Matrix

In contrast to a singular matrix, a nonsingular matrix has a determinant that is not zero. This kind of matrix is also known as an invertible or regular matrix.
Nonsingular matrices are critical in linear algebra since they allow us to perform more extensive computations and solve systems of equations effectively.

• Non-Zero Determinant: If a matrix's determinant is not zero, it confirms the matrix is nonsingular.
• Invertible: These matrices have inverses, which allow them to be used in various mathematical operations, like diagonalization or solving linear equations.

The ability to find an inverse is crucial because it provides a pathway to find solutions to matrix equations and helps in understanding more about the space that the matrix operates within.

Determinant of a Matrix

The determinant of a matrix is a scalar value that represents the volume distortion during the transformation described by the matrix. It is calculated using a specific formula based on the size and elements of the matrix.
For a 3x3 matrix, the determinant calculation involves using three different terms: a primary diagonal, two secondary diagonals, and applying specific operations.
Here's a brief breakdown on how to calculate it:

• Formula: For a matrix \(\begin{pmatrix}a & b & c \d & e & f \g & h & i \end{pmatrix}\), use the formula: \[Determinant = a(ei - fh) - b(di - fg) + c(dh - eg)\]
• Zero Determinant: If the result is zero, the matrix is singular. If not zero, it is nonsingular.
• Interpretation: The determinant helps in understanding the matrix's properties, such as invertibility and the geometric interpretations like area scaling for 2x2 matrices or volume scaling for 3x3 matrices.

Understanding how and why the determinant works as it does is a crucial part of getting comfortable with matrices in linear algebra.