Question

Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{cccc}0 & 0 & -1 & -1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & -1 & 0 \\\ -1 & 0 & 1 & 0\end{array}\right]\)

Short answer

The determinant of the matrix is -2.

Step-by-step solution

Identify the Matrix and Set Up the Determinant

The given matrix is:\[ A = \begin{bmatrix} 0 & 0 & -1 & -1 \ 1 & 1 & 0 & 1 \ 1 & 1 & -1 & 0 \ -1 & 0 & 1 & 0 \end{bmatrix} \] We need to calculate the determinant of this 4x4 matrix using cofactor expansion along the first row.

Set Up the Cofactor Expansion

For cofactor expansion along the first row, use the formula:\[\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} + a_{14}C_{14}\]where \(a_{ij}\) is the element in the \(i\)-th row and \(j\)-th column, and \(C_{ij}\) is the cofactor of \(a_{ij}\).

Calculate the Cofactors for the First Row

Cofactors are calculated using:\[ C_{ij} = (-1)^{i+j}M_{ij} \]where \( M_{ij} \) is the minor of the element \( a_{ij} \), obtained by removing the \(i\)-th row and \(j\)-th column.

Compute the Cofactor for a_{13}

- Element \( a_{13} = -1 \).- Minor matrix for \( a_{13} \):\[\begin{bmatrix}1 & 1 & 1 \1 & 1 & 0 \-1 & 0 & 0\end{bmatrix}\]- Compute the determinant of the 3x3 matrix using smaller cofactor expansion:\[M_{13} = (1) \begin{vmatrix} 1 & 0 \ 0 & 0 \end{vmatrix} - (1) \begin{vmatrix} 1 & 0 \ -1 & 0 \end{vmatrix} + (1) \begin{vmatrix} 1 & 1 \ -1 & 0 \end{vmatrix} = (1)(0) - (0) + (0 - (-1)(1)) = 1\]So, cofactor: \( C_{13} = (-1)^{1+3}(-1) = 1 \).

Compute the Cofactor for a_{14}

- Element \( a_{14} = -1 \).- Minor matrix for \( a_{14} \):\[\begin{bmatrix}1 & 1 & 0 \1 & 1 & -1 \-1 & 0 & 1\end{bmatrix}\]- Compute its determinant:\[M_{14} = (1)\begin{vmatrix} 1 & -1 \ 0 & 1 \end{vmatrix} - (1)\begin{vmatrix} 1 & -1 \ -1 & 1 \end{vmatrix} + (0) \begin{vmatrix} 1 & 1 \ -1 & 0 \end{vmatrix} = 1(1 + 0) - (1 \cdot 1 - (-1)\cdot 1) = 1 - 0 = 1\]So, cofactor: \( C_{14} = (-1)^{1+4}(1) = 1 \).

Perform the Cofactor Expansion

Substitute the calculated cofactors back into the cofactor expansion formula:\[ \det(A) = 0 \times C_{11} + 0 \times C_{12} + (-1) \times 1 + (-1) \times 1 \]\[\det(A) = -1 - 1 = -2\]

Conclusion: Determine the Determinant

The determinant of the matrix using cofactor expansion along the first row is \( -2 \).

Key concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a technique used to calculate the determinant of a matrix. The key idea is to break down a larger matrix into smaller parts, making the computation more manageable.

In the process of finding a determinant using cofactor expansion, you usually choose one row or column and "expand" across it. This means you sum up the products of each element in the row or column by their corresponding cofactors.

Here's the step-by-step approach on how it works:

• Select a row or column for expansion, often the first row or one with many zeros to minimize calculations.
• For each element in that row or column, calculate the corresponding cofactor, which involves a minor matrix and the sign adjustment factor \((-1)^{i+j}\).
• Multiply each element by its cofactor, and sum these products to get the determinant.

In our example matrix, cofactor expansion along the first row was applied to find its determinant. Remember, choosing a row or column with more zeros can simplify the computation significantly.

Matrix Algebra

Matrix algebra is a branch of mathematics focused on the study and manipulation of matrices. These are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns.

Understanding how to handle matrices is crucial because they represent systems of equations, transformations, and other mathematical phenomena in a concise form.

• Basic operations include addition, subtraction, and multiplication of matrices.
• Matrix multiplication can represent compositions of linear transformations.
• The determinant is a scalar value that reflects certain properties of a matrix, such as whether it is invertible.

In matrix algebra, the determinant of a matrix helps in analyzing the matrix's properties, like its invertibility. If the determinant of a square matrix is zero, it implies that the matrix does not have an inverse, which is crucial for solving systems of linear equations and other applications.

4x4 Matrix

4x4 matrices are matrices with four rows and four columns. These matrices are particularly important in various fields, including computer graphics, physics, and engineering.

Working with 4x4 matrices can be complex due to the large number of elements involved, requiring more computation time and effort. However, they provide a powerful way to perform transformations in four-dimensional space.

• They can represent transformations, rotations, and scaling in 3D space.
• Cofactor expansion lets you calculate the determinant step-by-step, even for such large matrices.
• Understanding 4x4 matrices is essential for advanced applications in various scientific fields.

In the context of solving the determinant, as seen in the original problem, following the structured process of cofactor expansion becomes vital to manage the complexity introduced by the 4x4 configuration.

Minor Matrix

The minor matrix of an element in a matrix is obtained by deleting the row and column that contain that element.

This minor matrix is crucial when calculating a cofactor for use in cofactor expansion, as it forms the basis for these calculations.

• To find the minor of an element, remove its row and column.
• Calculate the determinant of the resulting smaller matrix.
• This value is then adjusted with the sign factor \((-1)^{i+j}\) to form the cofactor.

In the original exercise, for instance, when computing the determinant through cofactor expansion, minor matrices simplify the larger matrix into smaller, more manageable parts, making it easier to determine the determinant of a 4x4 matrix.