Problem 42 - Differential Equations and Linear Algebra

Question

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrrrr}0 & 0 & 0 & 8 & 4 \\ 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 2 & 0 & 0 \\ 2 & -3 & 0 & 0 & 0 \\ 4 & -2 & 0 & 0 & 0\end{array}\right]\).

Step-by-Step Solution

Answer

The determinant of the given \(5\times5\) matrix A can be calculated using the cofactor expansion method. We expand along the first column since there are three zeros, minimizing calculations. After finding the cofactors and evaluating the \(4\times4\) determinants, we find that the determinant of matrix A is 32.

Step 1: Reminder of Formula

Let us do a quick revision of the cofactor expansion formula. For a \(5\times5\) matrix A, we can find its determinant by expanding along any row or column. In this case, we will expand along the first column. The determinant of a matrix A can be calculated as: \[ \text{det}(A) = \sum_{i=1}^5 (-1)^{i+1} a_{i1}C_{i1} \] where \(a_{i1}\) are the elements of the first column and \(C_{i1}\) are their corresponding cofactors.

Step 2: Calculate Each Cofactor

Now, we will calculate the cofactor for each non-zero element of the first column: 1. \(C_{41}\): Remove the 4th row and 1st column from A, and then find the determinant of the remaining \(4\times4\) matrix \[C_{41} = \text{det}\left(\begin{array}{rrrr} 0 & 0 & 8 & 4 \\ 0 & 0 & -1 & 1 \\ 0 & 2 & 0 & 0 \\ 4 & -2 & 0 & 0\end{array}\right)\] 2. \(C_{51}\): Remove the 5th row and 1st column from A, and then find the determinant of the remaining \(4\times4\) matrix \[C_{51} = \text{det}\left(\begin{array}{rrrr} 0 & 0 & 8 & 4 \\ 0 & 0 & -1 & 1 \\ 0 & 2 & 0 & 0 \\ 2 & -3 & 0 & 0\end{array}\right)\]

Step 3: Calculate the Determinant using Cofactor Expansion

Use the cofactor expansion formula to calculate the determinant of A: \[ \text{det}(A) = (-1)^{4+1}(2)C_{41}+(-1)^{5+1}(4)C_{51} \] Now, substitute the cofactors from Step 2: \[\text{det}(A) = (-1)^5(2)\text{det}\left(\begin{array}{rrrr} 0 & 0 & 8 & 4 \\ 0 & 0 & -1 & 1 \\ 0 & 2 & 0 & 0 \\ 4 & -2 & 0 & 0\end{array}\right) + (-1)^6(4)\text{det}\left(\begin{array}{rrrr} 0 & 0 & 8 & 4 \\ 0 & 0 & -1 & 1 \\ 0 & 2 & 0 & 0 \\ 2 & -3 & 0 & 0\end{array}\right) \]

Step 4: Evaluate Each 4x4 Determinant

Now, we can use the method of cofactor expansion again for the \(4\times4\) matrices. We will expand along the first column to speed up the process: \[ C_{41} = (-1)^6(2)\text{det}\left(\begin{array}{rrr} 0 & -1 & 1 \\ 2 & 0 & 0 \\ -2 & 0 & 0\end{array}\right) \] \[ C_{51} = (-1)^5(2)\text{det}\left(\begin{array}{rrr} 0 & 8 & 4 \\ 0 & -1 & 1 \\ 2 & 0 & 0\end{array}\right) \] Now, we will evaluate the determinants: \[C_{41} = (2)\text{det}\left(\begin{array}{rrr} 0 & -1 & 1 \\ 2 & 0 & 0 \\ -2 & 0 & 0\end{array}\right) = (2)(0)\] \[C_{51} = (-2)\text{det}\left(\begin{array}{rrr} 0 & 8 & 4 \\ 0 & -1 & 1 \\ 2 & 0 & 0\end{array}\right) = (-2)(8)\]

Step 5: Solve for the Determinant of A

Finally, substitute the values of \(C_{41}\) and \(C_{51}\) back into the formula from Step 3: \[ \text{det}(A) = (-1)^5(2)C_{41}+(-1)^6(4)C_{51} = (-1)^5(2)(0) + (-1)^6(4)(-8) = 32 \] So, the determinant of matrix A is 32.

Key Concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method used to calculate the determinant of a matrix. It's particularly useful for larger matrices like a 5x5 matrix. The process involves selecting a row or a column and then expanding the determinant into smaller matrices, which are easier to solve.

For a matrix, the cofactor expansion formula is:

• Choose a row or a column to expand. Usually, the row or column with the most zeros is optimal.
• Calculate the cofactor for each element of the chosen row or column. The cofactor is determined by removing the row and column of the element and finding the determinant of the remaining matrix. Don’t forget to consider the position’s sign, which alternates according to the checkerboard pattern.

The determinant is then written as a sum of these cofactors, multiplied by their respective elements in the row or column selected for expansion.

Linear Algebra

Linear Algebra is a branch of mathematics that focuses on the study of vectors, vector spaces, linear transformations, and systems of linear equations.

One of the fundamental aspects of linear algebra is the concept of matrices and operations on them, which includes finding determinants. Determinants give us crucial information about the matrix, such as:

• The volume scaling factor of the linear transformation described by the matrix.
• Whether the matrix is invertible. A non-zero determinant indicates the matrix has an inverse.
• The linear independence of rows or columns. If the determinant is zero, the rows or columns are linearly dependent.

Understanding how determinants work in linear algebra helps solve complex problems related to network analysis, physics simulations, and computer graphics.

5x5 Matrix Determinants

Calculating the determinant of a 5x5 matrix involves breaking it down into smaller, more manageable parts using cofactor expansion. This method is necessary because directly solving a 5x5 matrix's determinant is complex and time-consuming.

The procedure typically involves the following steps:

• Choose a row or column to expand, preferably one with zeros to simplify calculations.
• Perform cofactor expansion to reduce the 5x5 matrix into a series of smaller matrices, usually 4x4, and further reduce as necessary.
• Continue reducing the matrices to find their determinants, ultimately combining these back to find the original determinant.

This approach highlights how mathematical techniques can simplify complex problems, allowing for step-by-step resolution of large matrix determinants.