Problem 17 - Advanced Engineering Mathematics

Question

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} 3 & 0 & 2 \\ 2 & 7 & 1 \\ 2 & 6 & 4 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix is 46.

Step 1: Understand the Cofactor Expansion Formula

For a 3x3 matrix, the determinant using cofactor expansion along the first row is given by:\[det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\]where \(a_{ij}\) are the elements of the first row, and \(C_{ij}\) are the cofactors of those elements.

Step 2: Determine the First Row Elements and Their Cofactors

The elements in the first row of the matrix are 3, 0, and 2. We need to find the cofactors of these elements:- The cofactor \(C_{11}\) involves the submatrix \(\left(\begin{array}{ll} 7 & 1 \ 6 & 4 \end{array}\right)\).- The cofactor \(C_{12}\) involves the submatrix \(\left(\begin{array}{ll} 2 & 1 \ 2 & 4 \end{array}\right)\).- The cofactor \(C_{13}\) involves the submatrix \(\left(\begin{array}{ll} 2 & 7 \ 2 & 6 \end{array}\right)\).

Step 3: Calculate the Cofactors

The cofactor \(C_{ij}\) is calculated as \((-1)^{i+j}\) multiplied by the determinant of the submatrix.- \(C_{11} = (-1)^{1+1}(7\cdot4 - 1\cdot6) = 22\).- \(C_{12} = (-1)^{1+2}(2\cdot4 - 1\cdot2) = -6\).- \(C_{13} = (-1)^{1+3}(2\cdot6 - 7\cdot2) = -10\).

Step 4: Perform the Cofactor Expansion

Plug the cofactors into the expansion formula:\[det(A) = 3\times22 + 0\times(-6) + 2\times(-10) = 66 + 0 - 20 = 46\]

Step 5: Conclusion

The determinant of the matrix is calculated as 46 using cofactor expansion along the first row.

Key Concepts

Cofactor Expansion

Cofactor expansion is a method to compute the determinant of a matrix by expanding along a row or column. For a 3x3 matrix, this involves breaking down the large matrix into smaller 2x2 submatrices called minors. Here's how it works:- Choose a row or a column to expand along. Typically, it's easier along a row or column with zeros as it simplifies the calculation.- For each element in the chosen row or column, compute its cofactor. The cofactor is the determinant of the submatrix formed by removing the row and column of that element, multiplied by \((-1)^{i+j}\) where \((i,j)\) are the element's positions.- Multiply each element by its cofactor and sum these products to find the determinant.The beauty of cofactor expansion is that it breaks down the complex task of calculating determinants into more manageable steps that involve only 2x2 matrices.

3x3 Matrix

A 3x3 matrix is simply a square array consisting of three rows and three columns. Each entry in the matrix is identified by its position, typically given by two indices.- The matrix generally looks like this: \[ \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \ \end{bmatrix} \]- These matrices play a vital role in linear algebra, appearing frequently in transformations, solving systems of equations, and more.- Their determinant, a single value, gives insight into the matrix's properties, like whether it's invertible.The process of finding a determinant gives us a powerful tool to analyze systems represented by these matrices.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It underpins much of modern mathematics and has broad applications. - **Operations and Concepts**: Involves operations like addition, scalar multiplication, and finding inverses. Concepts like determinants, eigenvalues, and eigenvectors are central. - **Applicability**: Used in diverse fields such as computer graphics, engineering, machine learning, and physics. - **Matrix Theory**: The study of matrices is a fundamental part of linear algebra. Matrices represent linear transformations, which are movements within vector spaces. By understanding linear algebra, we gain insight into structures and transformations in both theoretical and applied contexts. It provides the mathematical foundation necessary for tackling complex problems in many scientific and engineering domains.