Question

Find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. \(\left[\begin{array}{rrrr}6 & 0 & -3 & 5 \\ 4 & 13 & 6 & -8 \\ -1 & 0 & 7 & 4 \\ 8 & 6 & 0 & 2\end{array}\right]\) (a) Row 2 (b) Column 2

Short answer

The determinant obtained from expansion by cofactors, either by using the second row or second column, should be the same. The exact numerical value will depend on the specific elements of the given matrix.

Step-by-step solution

Calculating Determinant by Expanding Row 2

Choosing the second row for expansion, the determinant is given by: \( det(A) = a_{21}C_{21} - a_{22}C_{22} + a_{23}C_{23} - a_{24}C_{24} \). Substitute the given values of second row into the formula and calculate the determinant. \( C_{ij} \) represents the cofactor of element at position (i,j).

Calculating Determinant by Expanding Column 2

Now, choose the second column for expansion. Just like the previous case, the determinant is calculated using: \( det(A) = a_{12}C_{12} - a_{22}C_{22} + a_{32}C_{32} - a_{42}C_{42} \). Substitute the given values of second column into the formula and calculate the determinant.

Verify Determinants

Regardless of which row or column is chosen for cofactor expansion, the determinant of the matrix should remain the same.

Key concepts

Expansion by Cofactors

The method of expansion by cofactors is a technique used for calculating determinants, particularly useful for larger matrices. This approach involves selecting a single row or column from the matrix and using its elements along with their corresponding cofactors to calculate the determinant. A cofactor, denoted as \(C_{ij}\), is calculated by taking the determinant of the submatrix that remains after removing the element's row and column, and then applying a sign change based on the element's position. The sign is positive if the sum of the indices is even, and negative if it's odd.

For instance, when expanding the determinant along row 2 or column 2, each element in the chosen row or column is multiplied by its respective cofactor and summed, following an alternating sign pattern. This sum yields the overall determinant of the matrix. Critically, the determinant will be the same regardless of which row or column is chosen, showcasing the consistent nature of this method.

Calculating Determinants

Determinants are a fundamental component in matrix algebra, with numerous applications including solving systems of linear equations and evaluating the invertibility of matrices. The determinant of a square matrix provides insight into whether the matrix is singular (noninvertible) or nonsingular (invertible).

To calculate the determinant of a matrix, one can use a variety of methods. For a 2x2 matrix, the determinant is found simply by crossing multiplying the elements and subtracting the products. However, for larger matrices, methods like expansion by cofactors become more practical. Determinants also have properties that can simplify computations, such as the fact that exchanging two rows or columns alters the sign of the determinant, and that matrices with a row or column of zeros have a determinant of zero. Understanding these properties can greatly aid in calculating determinants efficiently.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and their operations. Matrices, which are rectangular arrays of numbers, symbols, or expressions, are essential tools in various areas of both pure and applied mathematics. Basic operations include addition, subtraction, and multiplication of matrices, while more advanced operations involve finding inverses, determinants, and eigenvalues.

In matrix algebra, the determinant is crucial because it provides important properties about the matrix. For example, only square matrices (matrices with the same number of rows and columns) have determinants, and the determinant helps determine if a matrix is invertible. Additionally, certain matrix transformations and solving systems of linear equations often involve calculating the determinant at some stage.

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method used to compute the determinant of a square matrix. It is particularly useful for dealing with larger matrices where direct computation is not feasible. The essence of cofactor expansion is to break down a large determinant into smaller, more manageable parts by focusing on one row or column at a time.

The process involves multiplying each element of the chosen row or column by its corresponding cofactor and alternating the sign. The cofactors reflect the minors of the matrix, which are determinants of smaller matrices that result from eliminating the current element's row and column from the original matrix. Cofactor expansion leverages the recursive nature of determinants, permitting it to be applied repeatedly until reaching a determinant of a 2x2 matrix, which is straightforward to compute. Understanding and applying cofactor expansion is a critical skill for anyone working with higher-order determinants in matrix algebra.