Problem 22 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. \(\left[\begin{array}{llll}{1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {1} \\\ {1} & {1} & {1} & {0} \\ {1} & {1} & {1} & {1}\end{array}\right]\)

Step-by-Step Solution

Answer

The matrix does not have an inverse because its determinant is zero.

Step 1: Calculate the determinant

To find the inverse of a matrix, the matrix must be square, and its determinant should not be zero. For a 4x4 matrix like \(A\), the determinant can be computed using more advanced techniques, but in simpler cases, we can use known methods or calculator assistance to directly evaluate the determinant. The key observation here is that if the determinant is zero, the matrix does not have an inverse.

Step 2: Using Properties of Row Operations

Notice that row operations won't change the determinant if done correctly. Observe the structure of the given matrix:\[A = \begin{bmatrix} 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 \ 1 & 1 & 1 & 0 \ 1 & 1 & 1 & 1 \end{bmatrix}\]By performing some row operations (like subtracting row operations), recognize the rows are not linearly independent (e.g., Row 4 is Row 3 plus Row 1). Thus, the determinant is zero.

Step 3: Conclusion from Determinant Evaluation

Since we determined that the determinant is zero, the matrix is singular. A singular matrix does not have an inverse.

Key Concepts

Determinant

When working with matrices, the determinant is an essential concept, especially when trying to find the inverse. The determinant is a specific number that can be calculated from a square matrix. It provides important information about the matrix, such as whether it is invertible or not. If a matrix has a determinant of zero, it means the matrix is singular and does not have an inverse.
For a 4x4 matrix, calculating the determinant can be more complex. It often involves breaking the matrix down into smaller components or using advanced techniques. In many cases, the simpler methods used for 2x2 or 3x3 matrices do not directly apply, and assistance from computational tools or a structured algorithm may be needed.
In the exercise, the determinant was calculated to be zero, indicating that the matrix of interest does not have an inverse. Thus, knowing how to determine the determinant is a crucial skill for anyone working with matrices.

Row Operations

Row operations are procedures used to simplify matrices and solve linear equations. They include three main types of operations: swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row to another row. These operations are key tools when working with matrices, especially for finding the inverse or solving systems of equations.
Importantly, row operations do not change the determinant of a matrix, as long as they are done in ways that preserve certain properties:

• Swapping two rows will change the sign of the determinant.
• Multiplying a row by a scalar multiplies the determinant by that scalar.
• Adding a multiple of a row to another row does not change the determinant.

In the given exercise, by examining the row operations, it was observed that the rows were linearly dependent, further confirming the determinant is zero. This observation came from realizing that one row could be written as a combination of others, which means they are not independent. This type of analysis is helpful in understanding the structure of the matrix and its properties.

Singular Matrix

A matrix is termed "singular" if it does not have an inverse, and this occurs precisely when the determinant is zero. Singular matrices have particular characteristics that make them unique: for instance, they lead to systems of equations with either no solutions or infinitely many solutions, rather than a unique solution.
This happens because a determinant of zero implies that the matrix's rows or columns are not linearly independent, meaning at least one row (or column) can be expressed as a combination of others. This lack of independence prevents the matrix from being invertible.
Understanding singular matrices is important for applications that require matrix inversion. Recognizing that a matrix is singular precludes the need for attempting to find an inverse, saving time and guiding the solution path in mathematical problems and real-world applications.