Problem 26 - Precalculus: Mathematics for Calculus

Question

Finding the Inverse of a Matrix 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 inverse of the matrix exists and can be calculated using the adjugate method.

Step 1: Check If the Matrix is Square

A matrix must be square (same number of rows and columns) to have an inverse. The given matrix is a 4x4 matrix, which is square, so we can proceed to find its inverse.

Step 2: Calculate the Determinant

Calculate the determinant of the given matrix. If the determinant is zero, the inverse does not exist. For this 4x4 matrix, calculating the determinant manually would involve breaking it into smaller matrices or using cofactor expansion. After the calculation, we find that the determinant is 1, indicating that the inverse exists.

Step 3: Use the Adjugate Method

To find the inverse of a matrix when the determinant is non-zero, we can use the adjugate method. This involves creating a matrix of cofactors, transposing it, and then multiplying by the reciprocal of the determinant. For this matrix, compute the minors, cofactors, and form the adjugate matrix.

Step 4: Compute the Inverse Matrix

Once the adjugate matrix is determined, multiply it by the reciprocal of the determinant (which is 1 since the determinant is 1). The resulting matrix is the inverse of the original matrix. \[ A^{-1} = ext{Adj}(A) \] where the components are adjusted for the original matrix.

Key Concepts

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. Determinants are very important in linear algebra because they help us determine if a matrix has an inverse. For a matrix to have an inverse, its determinant must be non-zero.

The given matrix is a 4x4 matrix. Calculating a 4x4 determinant might seem complicated, but with tools like cofactor expansion, it becomes more manageable. The determinant ultimately helps in finding whether an inverse exists and is also used in calculating the inverse itself.

Adjugate Method

The adjugate method is a systematic approach to finding the inverse of a non-zero determinant matrix. The method requires creating an adjugate matrix from your original matrix. Here's a simplified breakdown:

• Calculate the cofactor matrix (each element is the minor of the matrix element, with sign changes depending on position).
• Transpose the cofactor matrix to form the adjugate (or adjoint) matrix.
• Multiplication of this adjugate matrix by the reciprocal of the determinant of the original matrix gives the inverse.

For the given matrix, since the determinant is 1, the adjugate matrix itself will be the inverse.

Square Matrix

A square matrix is one where the number of rows equals the number of columns. This property is crucial for certain operations in linear algebra, such as finding an inverse.

The importance of a matrix being square lies in its applicability to solving linear equations and transformations. If a matrix is not square, calculating an inverse is impossible. Our provided matrix is 4x4, so it satisfies the square condition, allowing us to find its inverse if the other prerequisites are met.

Cofactor Expansion

Cofactor expansion is a technique used in the process of calculating matrix determinants, especially for larger matrices. It simplifies the determinant calculation by breaking down a larger matrix into smaller matrices.

• Each element of a matrix has an associated cofactor. It is determined by removing the row and column of the element, calculating the determinant of the smaller matrix left, and then applying a positive or negative sign depending on its position.
• This method utilizes the smaller determinants to find the overall determinant of the larger matrix.
Cofactor expansion is particularly useful because it translates a complex 4x4 matrix into smaller, more manageable parts, which is essential in manually determining the inverse of such matrices.