Problem 36 - Precalculus

Question

Evaluate each \(3 \times 3\) determinant. $$\left|\begin{array}{rrr} -3 & 2 & -5 \\ 1 & 8 & 2 \\ 4 & -6 & 9 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is -64.

Step 1: Identify First Row Elements

We start by identifying the elements of the first row of the matrix, which are \(-3, 2,\) and \(-5\).

Step 2: Calculate Minor Determinants

Next, we calculate the determinant of each minor, which are formed by removing the row and column of each element from the first row:- For \(-3\):\[egin{vmatrix}8 & 2 \-6 & 9 ight\\right|= (8 \times 9) - (2 \times (-6)) = 72 + 12 = 84\]- For \(2\):\[egin{vmatrix}1 & 2 \4 & 9 ight\\right|= (1 \times 9) - (2 \times 4) = 9 - 8 = 1\]- For \(-5\):\[egin{vmatrix}1 & 8 \4 & -6 ight\\right|= (1 \times -6) - (8 \times 4) = -6 - 32 = -38\]

Step 3: Apply Cofactor Expansion

We use the formula for the determinant of a \(3 \times 3\) matrix using the first row elements:\[-3 imes 84 - 2 imes 1 - 5 imes (-38)\]Calculate each term:\[-3 imes 84 = -252, \2 imes 1 = 2, \-5 imes (-38) = 190\]

Step 4: Sum the Results

Add up all the results from the cofactor expansion:\[-252 - 2 + 190 = -64\]Therefore, the determinant is \(-64\).

Key Concepts

Cofactor Expansion

Cofactor expansion is a technique used in matrix algebra to calculate the determinant of a matrix, especially useful for matrices larger than 2x2. To perform cofactor expansion, we usually select a row or a column and calculate the cofactor for each element in that row or column.

The cofactor is calculated by multiplying the element with its minor and then adjusting for sign based on the position of the element. The sign is determined by the position by checking the combination of row and column indices. For a matrix element located at row i and column j, the sign will be definitely (+) if (i + j) is even and (-) if (i + j) is odd.

The formula for cofactor expansion along the first row in a 3x3 matrix looks like this:

• Element * Minor Matrix Determinant
• Sign Adjustment (+ or -)

Sum these modified products together to find the determinant. This method is particularly effective in determining the determinant of larger matrices.

Minor Determinants

Minor determinants are smaller determinants of a matrix obtained by deleting one row and one column from the original matrix. For each element of a matrix, a minor is formed.

For instance, consider calculating the determinant of a 3x3 matrix element as part of the first row. Remove the respective row and column of the element to get a 2x2 matrix. Then, calculate the determinant of this smaller matrix.

The calculation for a 2x2 determinant from the matrix \(\begin{bmatrix}a & b \ c & d\end{bmatrix}\) is simple:

• Multiply diagonally: \(ad - bc\)

Having these minor determinants is essential as they serve as building blocks in the cofactor expansion method, simplifying the complex calculation process down to simpler, manageable parts, which are easier to evaluate.

Matrix Algebra

Matrix algebra is a mathematical system for handling matrices that are used in numerous fields such as physics, engineering, computer science, and economics.

It involves operations like addition, subtraction, multiplication, and finding determinants or inverses. A key component of matrix algebra is understanding the structure and components of matrices, such as rows, columns, and elements. Here, we focus on determining the determinant, which is a scalar value reflecting certain properties of a square matrix.

The determinant is vital in solving linear equations, evaluating matrices in calculus, and determining whether a matrix is invertible.
Some important points about determinants include:

• They can indicate if a linear system has a unique solution (if the determinant is non-zero).
• Determinants change sign when two rows are swapped.

Understanding matrix algebra, particularly how to compute determinants, is crucial as it bridges many applications in mathematical models and real-world problem-solving scenarios.