Problem 31 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The determinant of the matrix, when expanded by Row 1 is -37 and by Column 3 is also -37. This further confirms the fact that the determinant of a matrix remains the same irrespective of the row or column we choose to expand by.

Step 1: Expand using Row 1

Take the first row of the matrix. The determinant of the matrix is the sum of the products of the elements of any row or column times their corresponding co-factors. Therefore, the determinant can be written as: det(A) = \(7*cof(1,1) + 0*cof(1,2) + -4*cof(1,3)\). The \(0*cof(1,2)\) term will go to zero because it is multiplying with zero.

Step 2: Calculate the co-factors for Row 1

The co-factor is defined as: (-1)^(i+j) * det(Minor(i,j)) where i,j is the position of the element in the matrix and det(Minor(i,j)) is the determinant of the matrix that remains after removing the i-th row and j-th column. Therefore, cof(1,1) = det(\[[-3,0],[8,1]\]) = -3, cof(1,3) = det(\[[2,0],[5,8]\]) = 16. Substitute these co-factors into the determinant expression obtained in Step 1.

Step 3: Calculate the determinant using Row 1

Plugging the cofactors into the expression, we get: det(A) = \(7*(-3) + -4*16 = -37\)

Step 4: Expand using Column 3

Take the third column of the matrix. The determinant of the matrix is the sum of the products of the elements of any row or column times their corresponding co-factors. Therefore, the determinant can be written as: det(A) = \(-4*cof(1,3) + 0*cof(2,3) + 1*cof(3,3)\). The \(0*cof(2,3)\) term will go to zero because it is multiplying with zero.

Step 5: Calculate the co-factors for Column 3

The cofactor for the elements of column 3 are: cof(1,3) = det(\[[2,-3],[5,8]\]) = 31, cof(3,3) = det(\[[7,0],[2,-3]\]) = -21. Substitute these co-factors into the determinant expression obtained in Step 4.

Step 6: Calculate the determinant using Column 3

Plugging the cofactors into the expression, we get: det(A) = \(-4*31 + 1*(-21) = -37\)

Key Concepts

Expansion by Cofactors

Expansion by cofactors is a powerful technique used to calculate the determinant of a square matrix. It works by expressing the determinant as a sum of products of elements and their corresponding cofactors. A cofactor is a signed minor of an element in the matrix. The cofactor for an element located at position (i,j) is calculated as \((-1)^{i+j} \times \text{det}(\text{Minor}(i,j))\), where \( (-1)^{i+j} \) provides the sign based on the position of the element, and \(\text{det}(\text{Minor}(i,j))\) is the determinant of the matrix obtained by deleting the ith row and jth column.

• When expanding, you can choose any row or column of the matrix. This provides flexibility in choosing the path of least computation.
• The expansion is especially useful for larger matrices or when zeros are present, as they simplify the calculations considerably.

Notice that if a row or column contains one or more zeros, the calculations become a bit easier because any term multiplying zero will vanish from the sum. This was evident in both step 1 using row 1 and step 4 using column 3 of our example.

Matrix Algebra

Matrix algebra is a fundamental part of linear algebra, involving the study of matrices and matrix operations such as addition, multiplication, and finding determinants. To correctly utilize expansion by cofactors, understanding the basics of matrix algebra is essential.

• Operations like addition and multiplication are governed by existing rules that ensure matrices behave predictably.
• The determinant is a special value that is computed from the elements of a square matrix providing insights such as invertibility, where a non-zero determinant indicates that the matrix is invertible.

In our exercise, the focus shifts onto determinants calculated using `expansion by cofactors`, reinforcing the importance of understanding matrix transformations and properties. This process demonstrates how determinants give meaningful insights into the properties of matrix equations.

Minor Determinants

Minor determinants are smaller determinants extracted from a larger square matrix. They form the basis for calculating cofactors, which in turn are critical for expanding a determinant by cofactors. Here's how to determine a minor:

• To calculate a minor determinant for a particular element, remove the row and column containing the element from the matrix.
• The remaining matrix constitutes the minor, and its determinant is referred to as the minor determinant.

For example, to find cof(1,1) in our problem, we remove the first row and first column, leaving the matrix \(\begin{bmatrix}-3 & 0 \ 8 & 1 \end{bmatrix}\). The determinant for this minor is the minor determinant that helps define the cofactor for the original position (1,1). Calculating these minor determinants correctly ensures accurate expansion and determinant calculation.