Problem 26 - Precalculus: Mathematical for Calculus

Question

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$\left|\begin{array}{rrrr} 2 & -1 & 6 & 4 \\ 7 & 2 & -2 & 5 \\ 4 & -2 & 10 & 8 \\ 6 & 1 & 1 & 4 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the matrix is \\(-400\\).

Step 1: Choose a Row or Column

To simplify our work, we should choose a row or column with the highest number of zeros or smallest entries. Observing the matrix, there are no zeros, but row 1 has the smallest numbers, so we will expand along row 1.

Step 2: Calculating Minor Determinants

We calculate the determinants needed for expanding the determinant along the first row using:\[det(A) = \begin{vmatrix}7 & 2 & 5 \4 & -2 & 8 \6 & 1 & 4\end{vmatrix}\] Use the cofactor expansion for entry (1,1), noting the sign and minor determinant:\[= 2 \det \begin{vmatrix} 2 & -2 & 5 \ -2 & 10 & 8 \ 1 & 1 & 4 \end{vmatrix} - (-1) \det \begin{vmatrix} 7 & -2 & 5 \ 4 & 10 & 8 \ 6 & 1 & 4 \end{vmatrix} + 6 \det \begin{vmatrix} 7 & 2 & 5 \ 4 & -2 & 8 \ 6 & 1 & 4 \end{vmatrix} - 4 \det \begin{vmatrix} 7 & 2 & -2 \ 4 & -2 & 10 \ 6 & 1 & 1 \end{vmatrix}\]

Step 3: Expand First Minor Determinant

Calculate the first minor determinant:\[= 2 \begin{vmatrix}2 & -2 & 5 \-2 & 10 & 8 \1 & 1 & 4\end{vmatrix}= 2((2(10)(4) + (-2)(8)(1) + 5(-2)(1)) - (1(10)(5) + 1(-2)(2) + 8(8)(2))) = 2(80 + 16 - 10 - 50 + 4 - 128)= 2(-88)= -176\]

Step 4: Calculate Second Minor Determinant

Next, we calculate the second minor determinant:\[= 1 \begin{vmatrix}7 & -2 & 5 \4 & 10 & 8 \6 & 1 & 4\end{vmatrix}= 1((7(10)(4) + (-2)(8)(6) + 5(1)(4)) - (4(10)(5) + 8(1)(7) + 1(-2)(4)))= 1(280 - 96 + 20 - 200 - 56 + 8)= -44\]

Step 5: Calculate Third Minor Determinant

Next, we calculate the third minor determinant:\[= 6 \begin{vmatrix}7 & 2 & 5 \4 & -2 & 8 \6 & 1 & 4\end{vmatrix}= 6(\det \begin{vmatrix} 2 & 5 \ -2 & 8 \end{vmatrix} - \det \begin{vmatrix} 4 & 8 \ 6 & 4 \end{vmatrix})\]Expand further:\[= 6((16 + 10) - (16 - 48)) = 6(26 + 32) = 6(58) = 348\]

Step 6: Calculate Fourth Minor Determinant

Calculate the fourth minor determinant:\[= 4 \begin{vmatrix}7 & 2 & -2 \4 & -2 & 10 \6 & 1 & 1\end{vmatrix}= 4(6(-2)(1) + 1(10)(7) + (-2)(-2)(6) - 1(-2)(4) - 10(7)(1) - 1(6)(-2)))= 4(-12 + 70 + 24 + 8 - 70 + 12)= 132\]

Step 7: Compile Determinant Result

Plug the computed minors back into the original expansion:\[det(A) = 2(-88) + 1(-44) + 6(58) - 4(132)= -176 - 44 + 348 - 528= -400\]

Step 8: Conclusion: Determine the Determinant Value

The determinant of the matrix is \(-400\), which we achieved by applying expansion by minors and simplifying using algebraic operations.

Key Concepts

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix by breaking it down into smaller components. This technique involves selecting a row or column of the matrix and using its elements to form a series of minor determinants. Each entry from the chosen row or column is multiplied by the determinant of the matrix obtained by deleting the row and column of that entry. Finally, these products are summed up, but not before adjusting each with a sign that alternates according to their position in the matrix.

• Start by identifying a row or column with the most zeros or smallest values to simplify the calculations.
• For each element in the chosen line, calculate its minor determinant.
• Apply a sign pattern (+, -, +, -) based on the element's position, starting from the top left of the matrix.

Cofactor expansion is particularly valuable in larger matrices where directly computing the determinant might be cumbersome. By dividing the problem into smaller, more manageable parts, the process becomes more systematic and computationally efficient. This technique enhances understanding of how determinants are structured and provides insight into their algebraic properties.

Minor Determinant

A minor determinant, often simply referred to as a "minor," is a key component in calculating a matrix's determinant through cofactor expansion. To find the minor of an element of a matrix, you delete the row and the column of that element, leaving a smaller matrix. The determinant of this smaller matrix is the minor.

For example, in our original matrix setup, to find the minor of the element in the first row and first column (2 from the number 2), you remove the first row and first column. This leaves you with a 3x3 matrix determinant that you need to evaluate.

• Remove the specific row and column connected to the chosen matrix element.
• Calculate the determinant of the reduced matrix to get the minor.

Minor determinants are fundamental in systematically reducing the complexity of determinant calculations. They allow you to focus on one segment of the matrix at a time, making big matrix calculations much more manageable. Minors also help reveal the deeper structural properties of matrices, which are often leveraged in other areas of mathematics and applied sciences.

Matrix Operations

Matrix operations are the foundational building blocks in solving problems in linear algebra, particularly when dealing with determinants. Matrix operations include a variety of actions, such as addition, subtraction, scalar multiplication, and, importantly, the operation of evaluating determinants.

When determining the determinant of a matrix, one often leverages various operations to simplify the process:

• Row and column operations can help reduce the complexity of the matrix, making it easier to calculate its determinant.
• Swapping two rows (or columns) multiplies the determinant by -1.
• Multiplying a row (or column) by a scalar multiplies the determinant by that same scalar.

Operationally, understanding how these manipulations affect the determinant is crucial. Simplifying a matrix with operations can lead to a more efficient calculation of its determinant, saving time and reducing errors. Mastery of matrix operations and their effects provide essential tools for advanced studies in algebra and calculus, enabling more sophisticated analyses and problem-solving capabilities.