Question

Find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. \(\left[\begin{array}{rrrr}10 & 8 & 3 & -7 \\ 4 & 0 & 5 & -6 \\ 0 & 3 & 2 & 7 \\\ 1 & 0 & -3 & 2\end{array}\right]\) (a) Row 3 (b) Column 1

Short answer

Without exact numerical results because the submatrix determinants were not provided, the determinant of the matrix expanding by Row 3 is 0*det(submatrix1) - 3*det(submatrix2) + 2*det(submatrix3) - 7*det(submatrix4) and by Column 1 is: 10*det(submatrix1) - 4*det(submatrix2) + 0*det(submatrix3) - 1*det(submatrix4).

Step-by-step solution

- Expansion of cofactors along Row 3

Each entry of the row is multiplied by its corresponding cofactor. The determinant then is the sum of these products. Calculating the cofactors involves determining the determinant of the sub-matrices that are left when the row and column of the chosen element are removed. This gives us 0*det(submatrix1) - 3*det(submatrix2) + 2*det(submatrix3) - 7*det(submatrix4). One must calculate the determinant of each submatrix accordingly.

- Expansion of cofactors along Column 1

Similar to step 1 but applied to the first column instead of the third row. The result is: 10*det(submatrix1) - 4*det(submatrix2) + 0*det(submatrix3) - 1*det(submatrix4). Now calculate the determinant of each submatrix accordingly.

Key concepts

Cofactor Expansion

To find the determinant of a matrix using cofactor expansion, one must think of it as a process of constructing smaller determinants from a larger matrix. Here's the idea: each element within the selected row or column is associated with a cofactor, which is pivotal in calculating the overall matrix determinant. A cofactor is a signed minor of the matrix obtained by deleting the row and column of the chosen element.

• Minor: A smaller matrix determinant obtained by removing one row and one column from the original matrix.
• Cofactor: It’s the minor determinant multiplied by (-1)^{i+j}, where "i" is the row number and "j" is the column number.

By doing the expansion (whether by a row or a column), you multiply each matrix entry by its corresponding cofactor, then add or subtract these products to obtain the determinant.

Row Expansion

Row expansion is a straightforward method to compute the determinant of a matrix. This approach involves selecting a specific row from the matrix. Let’s say you choose Row 3 from the matrix as per the exercise. Once a row is chosen, treat each of its elements as independent:

• Multiply each element by its cofactor.
• The sign alternates between positive and negative, starting with a positive from left to right.
• Add these algebraic products together.

For Row 3: - Start with 0 (the element in Row 3's first column), then move to 3, 2, and finally 7.- You calculate something like: \[ 0 \times ext{det(submatrix1)} - 3 \times ext{det(submatrix2)} + 2 \times ext{det(submatrix3)} - 7 \times ext{det(submatrix4)} \]By carrying out these operations, you need to compute the determinant of each smaller submatrix obtained after excluding the current row and column associated with each element.

Column Expansion

Using column expansion to compute a determinant is just as methodical as row expansion. In this case, you'll choose a specific column from the matrix to work with. For example, if Column 1 is selected:

• Each element of the column is multiplied by its associated cofactor.
• Alternate the signs between the positive and negative starting with positive.
• Add these products to find the determinant value.

For Column 1: - Work with 10, 4, 0, and 1 (elements in Column 1)- Compute it as: \[ 10 \times ext{det(submatrix1)} - 4 \times ext{det(submatrix2)} + 0 \times ext{det(submatrix3)} - 1 \times ext{det(submatrix4)} \]By adding these calculated products, and considering the alternating signs, you find the determinant of the entire matrix.

Submatrix Determinant

Whenever performing a cofactor expansion, a significant part entails finding submatrix determinants. A submatrix is obtained by eliminating one specific row and one specific column from the original matrix, correlating to the element being expanded. These are smaller matrices, often 2x2 or 3x3, depending on the size of the original. Calculating their determinants follows the standard formula:

• For a 2x2 matrix, use: \[ ext{det} = ad - bc \], where a, b, c, and d are the elements.
• For larger matrices, apply the same process of cofactor expansion recursively until you reach 2x2 matrices.

Submatrix determinants are the key to calculating the determinate of larger matrices using the cofactor expansion method. They bridge the gap between a large matrix and simple 2x2 interactions.