Problem 22 - Precalculus: Mathematical for Calculus

Question

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrrr} 1 & 2 & 0 & 2 \\ 3 & -4 & 0 & 4 \\ 0 & 1 & 6 & 0 \\ 1 & 0 & 2 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 56; thus, the matrix has an inverse.

Step 1: Identify the Matrix

We are given a 4x4 matrix: \[\begin{bmatrix}1 & 2 & 0 & 2 \3 & -4 & 0 & 4 \0 & 1 & 6 & 0 \1 & 0 & 2 & 0 \\end{bmatrix}\]We need to find the determinant of this matrix.

Step 2: Choose a Row or Column

For calculating the determinant of a 4x4 matrix, choose a row or column to expand along. Here, we'll choose the first row for simplification.

Step 3: Expand Along the First Row

Using the first row to expand the determinant, we have:\[det(A) = 1 \cdot \begin{vmatrix} -4 & 0 & 4 \ 1 & 6 & 0 \ 0 & 2 & 0 \end{vmatrix} - 2 \cdot \begin{vmatrix} 3 & 0 & 4 \ 0 & 6 & 0 \ 1 & 2 & 0 \end{vmatrix} + 0 \cdot \begin{vmatrix} 3 & -4 & 4 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{vmatrix} - 2 \cdot \begin{vmatrix} 3 & -4 & 0 \ 0 & 1 & 6 \ 1 & 0 & 2 \end{vmatrix}\]The terms involving multiplication by zero can be ignored.

Step 4: Calculate Each 3x3 Minor

Focus on the two non-zero minors:1. The first 3x3 minor for coefficient 1:\[\begin{vmatrix} -4 & 0 & 4 \ 1 & 6 & 0 \ 0 & 2 & 0 \end{vmatrix}\]Expand this minor along the first column:\[-4 \cdot (6 \cdot 0 - 0 \cdot 2) - 0 \cdot (1 \cdot 0 - 0 \cdot 2) + 4 \cdot (1 \cdot 2 - 6 \cdot 0)\]Resulting in:\[-4 \cdot 0 + 0 \cdot 0 + 4 \cdot 2 = 8\]2. The second 3x3 minor for coefficient -2:\[\begin{vmatrix} 3 & 0 & 4 \ 0 & 6 & 0 \ 1 & 2 & 0 \end{vmatrix}\]Expand this minor along the first column:\[3 \cdot (6 \cdot 0 - 0 \cdot 2) - 0 \cdot (0 \cdot 0 - 1 \cdot 2) + 4 \cdot (0 \cdot 2 - 6 \cdot 1)\]Resulting in:\[0 + 0 - 4 \cdot 6 = -24\]

Step 5: Calculate the Determinant

Combine the results from the two non-zero minors:\[det(A) = 1 \cdot 8 - 2 \cdot (-24) = 8 + 48 = 56\]

Step 6: Check if the Matrix is Invertible

Since the determinant of the matrix is not zero (\(det(A) = 56\)), the matrix is invertible.

Key Concepts

Matrix Inverse

Understanding whether a matrix has an inverse depends on its determinant. For a given matrix, if its determinant is non-zero, we can conclude that the matrix has an inverse.

In simpler terms, the ability to "reverse" or "invert" a matrix is linked to the determinant value.

• If the determinant is zero, the matrix is singular and does not have an inverse.
• If the determinant is non-zero, the matrix is invertible.

In our case, the 4x4 matrix provided has a determinant of 56. Since this value is not zero, we can confidently say that the matrix is invertible.

No need to calculate the inverse yet; identifying the determinant being non-zero is sufficient for confirming the existence of an inverse.

Matrix Determinant Calculation

The determinant of a matrix is a special number linked to a square matrix. It provides valuable information about the matrix, including whether it is invertible.

To calculate the determinant of a 4x4 matrix, you can use the process called cofactor expansion. This involves choosing any row or column, and using it to expand the determinant.

Let's summarize the steps involved:

• Choose a row or column to simplify the calculation process.
• Expand the determinant along the chosen row or column, developing smaller 3x3 matrices called minors.
• Calculate the determinant for each of these 3x3 matrices.
• Combine these determinants based on the original expansion formula to find the final determinant of the 4x4 matrix.

In our exercise, we selected the first row for expansion. This smart choice simplified our calculations considerably. The process led us through smaller determinants and helped us arrive at the final value of 56.

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a technique used to calculate the determinant of larger matrices. It systematically breaks down a matrix into smaller, more manageable pieces.

Choosing a row or column is the first step. The key is to select a row or column that simplifies calculations the most. In our example, expanding using the first row was strategic because it contained zeros, reducing the number of calculations.

The formula for a 4x4 determinant expansion with the first row is:

• For a given matrix \(A\), the determinant can be expanded as:
  \(det(A) = a_{11} \cdot C_{11} - a_{12} \cdot C_{12} + a_{13} \cdot C_{13} - a_{14} \cdot C_{14}\)
• Here \(a_{ij}\) are the elements of the selected row, and \(C_{ij}\) are their corresponding cofactors.

Remember:

• Cofactors involve computing the determinant of the minor, resulting from eliminating the row and column of the current element.
• Cofactors switch signs systematically based on their position.

In practice, this expansion method efficiently breaks the task into smaller determinants that are manageable and easy to compute.