Problem 4 - Advanced Engineering Mathematics

Question

To determine whether the given matrix is singular or nonsingular. $$ \left(\begin{array}{rr} 1 & -1 \\ 4 & 3 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is nonsingular.

Step 1: Understand the Problem

We need to determine if the given matrix is singular or nonsingular. A matrix is singular if its determinant is zero. Conversely, it is nonsingular if its determinant is non-zero.

Step 2: Recall the Formula for the Determinant of a 2x2 Matrix

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

Step 3: Substitute the Matrix Elements into the Determinant Formula

Substitute the elements from the matrix \( \begin{pmatrix} 1 & -1 \ 4 & 3 \end{pmatrix} \) into the determinant formula. Here, \( a = 1 \), \( b = -1 \), \( c = 4 \), and \( d = 3 \).

Step 4: Calculate the Determinant

Compute the determinant using the formula: \[ det = (1)(3) - (-1)(4) = 3 + 4 = 7. \]

Step 5: Draw the Conclusion

Since the determinant of the matrix is 7, which is not zero, the matrix is nonsingular.

Key Concepts

Determinant

The determinant is a calculated value that helps determine properties of a matrix, including whether a matrix is singular or nonsingular. For a 2x2 matrix, the determinant can be calculated using a simple formula. If the matrix is represented as \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \], then the formula to find the determinant is:

• \( det = ad - bc \)

The idea is to multiply the elements of the principal diagonal and subtract the product of the elements of the other diagonal. For our example:

• Given matrix: \( \begin{pmatrix} 1 & -1 \ 4 & 3 \end{pmatrix} \)
• Determinant: \( det = (1)(3) - (-1)(4) = 3 + 4 = 7 \)

Since the determinant is non-zero, further analysis of the matrix can be done showcasing its interesting properties.

Singular Matrix

A singular matrix is a matrix that does not have an inverse. This happens when the determinant of the matrix is zero. It is important in linear algebra because a singular matrix indicates that there are infinitely many solutions or no solution to a system of equations represented by the matrix.

• A singular matrix has a determinant of zero.
• Non-singular matrices have a non-zero determinant and therefore have an inverse.

Our example matrix, \[ \begin{pmatrix} 1 & -1 \ 4 & 3 \end{pmatrix} \], has a determinant of 7. As a result, it is a nonsingular matrix. This means there is a unique solution to its corresponding system of equations, and an inverse exists.

Matrix Analysis

Matrix analysis is a powerful tool used to study matrices by examining their properties and behaviors. Through this analysis, we learn how matrices can represent and solve linear equations, perform transformations, and rule over applications in various fields.

• Determinants are calculated to analyze whether matrices are invertible.
• Aid in solving systems of linear equations.
• Provide insights into linear transformations, such as rotations and scalings.

Knowing the determinant is just the beginning in matrix analysis. Understanding how each element contributes enables predictions regarding system behavior and transformation. Our evaluation of the determinant of \[ \begin{pmatrix} 1 & -1 \ 4 & 3 \end{pmatrix} \] helped us determine its nonsingular nature, showing that the matrix possesses a full rank and invertibility in matrix analysis.