Problem 24 - Precalculus Mathematics for Calculus

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: Check matrix invertibility

To find the inverse of a matrix, we first need to check if the matrix is invertible. A matrix is invertible if and only if its determinant is non-zero. Compute the determinant of the given 4x4 matrix using cofactor expansion.

Step 2: Determinant Calculation

Calculate the determinant of the matrix: \[\begin{vmatrix} 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 \ 1 & 1 & 1 & 0 \ 1 & 1 & 1 & 1 \end{vmatrix} \].Proceed with the cofactor expansion along the first row. The determinant simplifies to 0, since two rows are identical, which confirms that the matrix is not invertible.

Step 2: Conclusion about the inverse

Since the determinant of the matrix is 0, the matrix is not invertible. Therefore, it does not have an inverse.

Key Concepts

Matrix Determinant

The determinant is a special number that can be calculated from a square matrix. It's crucial because it helps determine the properties of a matrix. For a 4x4 matrix, the determinant captures essential information about the matrix's structure.
Simply put, if the determinant is zero, the matrix is singular and not invertible. But if the determinant is non-zero, the matrix can potentially have an inverse. Calculating the determinant involves summing up the products of certain elements of the matrix and their corresponding minors. This is often done using cofactor expansion, especially for higher-order matrices.
Here, when calculating the determinant of the given matrix, it becomes apparent that the determinant is zero because two rows are identical. This is a clear indicator that the matrix does not have an inverse.

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix. It involves choosing an entire row or column from the matrix and calculating the sum of its elements' products with their respective cofactors.
The cofactor of an element is the determinant of the submatrix that remains after removing the row and column of that element, each multiplied by \(-1\) raised to the sum of the element's row and column indices. This process is well-suited for calculating determinants of larger matrices when direct computation might be too complex.
In the given example, cofactor expansion along the first row was used to simplify the process of finding the determinant. Through this, the determinant was found to be zero, indicating potential issues with finding an inverse.

Invertibility of Matrices

The invertibility of a matrix is a fundamental concept in linear algebra. A matrix is considered invertible if there exists another matrix that, when multiplied with it, results in the identity matrix. However, not all matrices are invertible.
The key condition for invertibility is that the determinant of the matrix must be non-zero. If the determinant is zero, like in our example, the rows or columns of the matrix are linearly dependent. This means there is no unique inverse matrix that can reverse the effects of the original one.
In simple terms, a non-invertible matrix lacks a certain type of distinctiveness or uniqueness in its rows or columns, resulting in an absence of an inverse.