Question

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

Short answer

The determinant of the matrix is 0.

Step-by-step solution

Identify the Matrix

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

Setup Cofactor Expansion

To use the cofactor expansion along the first row, we will consider each element, multiply it by its corresponding cofactor, and sum these products. The formula is: \ \( \text{det}(A) = \sum_{j=1}^{n} (-1)^{1+j} \cdot a_{1j} \cdot \text{det}(A_{1j}) \) \Where \(A_{1j}\) is the submatrix after removing the first row and the \(j\)-th column for the element \(a_{1j}\).

Determine Cofactor for Each Element in First Row

1. For the element \(-1\) at position \((1,1)\), the corresponding cofactor is \(C_{11} = (-1)^{1+1} \cdot \text{det}(\begin{bmatrix}0 & 0 & 1 \ 1 & 1 & 0 \ 0 & -1 & -1 \end{bmatrix})\).2. For the element \(0\) at position \((1,2)\), the cofactor will be \(0\cdot\text{det}(\text{some matrix}) = 0\).3. For the element \(0\) at position \((1,3)\), the cofactor will also be \(0\).4. For the element \(-1\) at position \((1,4)\), the corresponding cofactor is \(C_{14} = (-1)^{1+4} \cdot \text{det}(\begin{bmatrix}-1 & 0 & 0 \ 1 & 1 & 1 \ 1 & 0 & -1 \end{bmatrix})\).

Calculate Determinant of Minor Matrices

For \(C_{11}:\)Using cofactor expansion along the first row of 3x3 matrix, calculate \[\text{det}\begin{bmatrix}0 & 0 & 1 \ 1 & 1 & 0 \ 0 & -1 & -1 \end{bmatrix} = 0(\text{det} \begin{bmatrix}1 & 0 \ -1 & -1 \end{bmatrix}) - 0 + 1(\text{det} \begin{bmatrix}1 & 1 \ 0 & -1 \end{bmatrix})\]\[ = 1(-1 - 0) = -1\]For \(C_{14}:\)Using cofactor expansion along the first row of 3x3 matrix:\[\text{det}\begin{bmatrix}-1 & 0 & 0 \ 1 & 1 & 1 \ 1 & 0 & -1 \end{bmatrix} = -1(\text{det} \begin{bmatrix}1 & 1 \ 0 & -1 \end{bmatrix}) + 0 + 0\]\[ = -1(-1 - 0) = 1\]

Calculate Determinant of the Original Matrix

To find the determinant of the original matrix, substitute the cofactor values found into the cofactor expansion:\[\text{det}(A) = (-1)1(-1) + 0 + 0 + (-1)1 = 1 - 1 = 0\]Thus, the determinant of the matrix is 0.

Key concepts

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix. The determinant is a special number that can be computed for square matrices, and it provides important information about the matrix. To perform cofactor expansion, you typically choose a row or a column of the matrix to expand along.
You will then, for each element in that row or column, calculate the cofactor, which is the minor of the element times \((-1)^{i+j}\) where \(i\) is the row index and \(j\) is the column index. This cofactor is then multiplied by the element itself. Finally, you sum these products to find the determinant. This method is especially useful for smaller matrices like 3x3 or 4x4 matrices, where direct computation isn't too cumbersome.

4x4 Matrix Determinant

Finding the determinant of a 4x4 matrix might seem intimidating at first. This is because unlike the easier 2x2 or 3x3 matrices, you have more elements to work with. However, using cofactor expansion simplifies the process.
Once you decide which row or column to expand along, usually the one with the most zeros to simplify calculations, you compute a smaller determinant for each remaining 3x3 minor matrix. Even though this process involves several steps, breaking it down makes it manageable. Each 3x3 determinant calculated then contributes to the final determinant of the 4x4 matrix.

Submatrix Calculation

When calculating determinants via cofactor expansion, you'll often need to find the determinant of a submatrix. A submatrix is formed by deleting certain rows and columns from a larger matrix.
This method allows you to tackle the main problem (finding the determinant of a larger matrix) by estimating determinants of smaller sections. To calculate a submatrix determinant from a 4x4 matrix, you typically create a 3x3 matrix by removing the row and column of the element you're focusing on. This step is crucial for determining cofactors and thus solving the determinant of larger matrices.

3x3 Matrix Determinant

The calculation of a 3x3 matrix determinant requires understanding the structure of smaller matrices. To find a 3x3 determinant, you can use a system similar to the cofactor expansion.
This entails choosing a row or column, computing partial products, and summing them. With a 3x3 matrix, you apply the following more manageable formula: \( ext{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\). You use the elements of the first row (a, b, c) and their corresponding minors to solve it. This formula makes it much quicker to find a determinant for 3x3 matrices compared to larger matrices.