Problem 5 - Advanced Engineering Mathematics

Question

To determine whether the given matrix is singular or nonsingular. $$ \left(\begin{array}{rr} 6 & 0 \\ -3 & 2 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is nonsingular.

Step 1: Understanding Singular Matrix

A matrix is considered singular if its determinant is 0. Conversely, if the determinant is non-zero, the matrix is nonsingular.

Step 2: Formula for Determinant of a 2x2 Matrix

For a 2x2 matrix of the form \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant \( \det(A) \) is calculated as: \[ \det(A) = ad - bc \]

Step 3: Plug Values into the Determinant Formula

Given the matrix \( \left( \begin{array}{rr} 6 & 0 \ -3 & 2 \end{array} \right) \), assign \( a=6 \), \( b=0 \), \( c=-3 \), and \( d=2 \). Substitute these values into the determinant formula: \[ \det(A) = (6)(2) - (0)(-3) \]

Step 4: Calculate the Determinant

Perform the calculation: \[ \det(A) = 12 - 0 = 12 \]

Step 5: Determine Matrix Singularity

Since the determinant is 12 and not zero, we conclude that the given matrix is nonsingular.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can tell us a lot about the matrix itself. For instance, it can reveal whether a matrix is singular or nonsingular. The determinant is calculated using specific mathematical formulas, which differ based on the size of the matrix. In general, the determinant helps to determine if the matrix has an inverse. Only nonsingular matrices, which have non-zero determinants, are invertible. Calculating the determinant is a crucial step in understanding the properties and characteristics of matrices in linear algebra.

2x2 Matrix Determinant

Calculating the determinant of a 2x2 matrix is straightforward. Consider a 2x2 matrix:\[\begin{pmatrix}a & b \c & d\end{pmatrix}\]To find the determinant, you use the formula:\[\det(A) = ad - bc\]This means you multiply the elements on the main diagonal (from top left to bottom right), \(a\) and \(d\), and subtract the product of the elements on the other diagonal, \(b\) and \(c\). If we take a specific example, say:\[\begin{pmatrix}6 & 0 \-3 & 2\end{pmatrix}\]By plugging these values into our determinant formula, \(6 \times 2 - 0 \times (-3)\), we'll see the determinant is 12.

Singular vs Nonsingular Matrix

The concept of singular and nonsingular matrices stems from the determinant value.

• A matrix is called singular if its determinant is zero.
• A nonsingular matrix, on the other hand, has a non-zero determinant.

Understanding the distinction is crucial for applications in mathematics and engineering, as nonsingular matrices have unique properties such as invertibility. In our example, the matrix:\[\begin{pmatrix}6 & 0 \-3 & 2\end{pmatrix}\]has a determinant of 12, which is not zero. Therefore, this matrix is concluded to be nonsingular, meaning it is invertible and does not lose any information when transformations are applied. This characteristic is essential in solving linear equations and many other mathematical problems.