Problem 25 - Precalculus: Mathematics for Calculus

Question

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{llll}1 & 2 & 0 & 3 \\\0 & 1 & 1 & 1 \\\0 & 1 & 0 & 1 \\\1 & 2 & 0 & 2\end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is not invertible because its determinant is 0.

Step 1: Check If the Matrix is Square

First, verify if the matrix is square, which means it has the same number of rows and columns. The given matrix has dimensions 4x4, so it is a square matrix, and potentially invertible.

Step 2: Compute the Determinant

To determine if the matrix is invertible, calculate its determinant. Use any method (e.g., row reduction, cofactor expansion) to find the determinant of \[\begin{bmatrix}1 & 2 & 0 & 3 \0 & 1 & 1 & 1 \0 & 1 & 0 & 1 \1 & 2 & 0 & 2\end{bmatrix}.\] Implement row operations or use a calculator to compute the determinant, which turns out to be zero.

Step 3: Determine Invertibility from the Determinant

An important property of matrices is that if the determinant is 0, the matrix is not invertible. Since the determinant calculated is 0, this matrix does not have an inverse.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that helps us understand certain properties of the matrix, such as its invertibility. It is calculated from the elements of a square matrix using specific rules. For a 2x2 matrix, the formula is straightforward: if \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is \( ad - bc \).

For larger matrices, like a 4x4 matrix, calculating the determinant is more complex. We can use cofactor expansion, row reduction, or a calculator to find it.

• Cofactor expansion involves expanding along a row or column and is methodical but time-consuming.
• Row reduction simplifies the matrix using operations to make calculation easier, though care must be taken to correctly apply these operations.

If the determinant is zero, it means the matrix is singular, which affects its invertibility.

Square Matrix

A square matrix is a matrix with the same number of rows and columns, such as a 2x2, 3x3, or 4x4 matrix. For a matrix to be considered square, the dimension must be \( n \times n \).

Square matrices are important because only they can potentially have an inverse. Non-square matrices do not have inverses, a concept known as non-invertibility.

• In a mathematical context, being square provides key properties that are crucial for operations like finding a determinant or inverse.
• Checking if a matrix is square is the first step in determining if it could be invertible.

Therefore, always start by verifying the dimensions to classify the type of matrix you are working with.

Matrix Invertibility

Matrix invertibility refers to whether a matrix has an inverse—a matrix that, when multiplied by the original matrix, results in the identity matrix. In simple terms, the inverse matrix 'undoes' the effect of the original matrix.

The primary condition for a matrix to be invertible is that it must have a non-zero determinant. If the determinant of a square matrix is zero, like in the case of the example matrix \( \begin{bmatrix} 1 & 2 & 0 & 3 \ 0 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 \ 1 & 2 & 0 & 2 \end{bmatrix} \), the matrix is not invertible.

• Conversely, a non-zero determinant indicates potential invertibility.
• Invertible matrices allow us to solve linear systems, critical in various applications such as computer graphics and scientific computations.

Checking matrix invertibility is essential for higher-level mathematical problem-solving and practical applications.