Problem 15 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} 0 & 2 & 0 \\ 3 & 0 & 1 \\ 0 & 5 & 8 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant is 48.

Step 1: Understand the Matrix

The matrix given is a 3x3 matrix:\[\begin{pmatrix}0 & 2 & 0 \3 & 0 & 1 \0 & 5 & 8 \\end{pmatrix}\]

Step 2: Select a Row or Column

For cofactor expansion, select a row or a column. Choosing one with zeros can simplify the calculations. Let's choose the first row: \(0\), \(2\), \(0\).

Step 3: Calculate Cofactor for Each Element

Calculate the cofactor expansion using the first row. Only the non-zero element is important:- For the element \(a_{12} = 2\), the cofactor is calculated by removing the first row and second column, resulting in the submatrix:\[\begin{pmatrix}3 & 1 \0 & 8 \\end{pmatrix}\]The determinant of this submatrix is:\(3 \times 8 - 1 \times 0 = 24\).

Step 4: Compute the Determinant using Cofactors

Using cofactor expansion along the first row, the determinant is:\(0 \times C_{11} + 2 \times C_{12} + 0 \times C_{13}\).Substitute the cofactor value:\( \text{determinant} = 2 \times 24 = 48\).

Key Concepts

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix. It involves breaking down a matrix into smaller matrices, known as minors, and then using them to compute the determinant. This method can significantly simplify the calculation, especially when the matrix contains zeros. The process begins by selecting any row or column of the matrix to expand along.

• An element from the chosen row or column is paired with a corresponding cofactor.
• To find a cofactor, ignore the current row and column of the element. The remaining elements form a smaller matrix.
• The determinant of this smaller matrix, adjusted by a sign based on the element's position, becomes the cofactor.

Multiply each element in the selected row or column by its cofactor and sum all these products to find the determinant. By using cofactor expansion effectively, the computational labor can be minimized, especially for matrices with numerous zeroes.

Matrix Algebra

Matrix algebra is a branch of mathematics that involves operations on matrices, which are rectangular arrays of numbers or functions. It is a fundamental concept with applications ranging from solving systems of linear equations to computer graphics and statistics.

• Common operations include addition, subtraction, and multiplication of matrices.
• Another important operation is calculating the determinant, which provides valuable properties about the matrix, such as singularity or invertibility.

To perform these operations, specific rules must be followed, such as matching dimensions for addition and using the dot product for multiplication. Determinants, particularly, are crucial as they can tell us if the matrix can be inverted or help in finding solutions to systems of equations. Mastery of these fundamental notions allows for advanced exploration into more complex mathematical and applied problems.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns. It is a commonly used size in various mathematical problems and applications. Calculating the determinant of a 3x3 matrix is a common exercise in linear algebra.

• One method to find the determinant involves the cofactor expansion discussed earlier.
• The arrangement of elements within a 3x3 matrix makes it relatively straightforward to apply these techniques.
• This process often becomes repetitive, reinforcing the understanding of minors and cofactors.

Understanding how to manipulate 3x3 matrices is essential for solving equations, transforming spaces, or even performing 3D graphics operations. As students practice with these matrices, they develop a deeper comprehension of both theoretical and practical applications, preparing them for more complex matrix manipulations.