Problem 16 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The determinant of the matrix is -30.

Step 1: Understand the Matrix

The matrix given is a 3x3 diagonal matrix: \[ A = \begin{pmatrix} 5 & 0 & 0 \ 0 & -3 & 0 \ 0 & 0 & 2 \end{pmatrix} \]A diagonal matrix has all non-zero elements on its main diagonal and zeros elsewhere.

Step 2: Recall Determinant of a Diagonal Matrix

The determinant of a diagonal matrix is the product of its diagonal elements. For a matrix:\[ \begin{pmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{pmatrix} \]the determinant is given by \( a \times b \times c \).

Step 3: Compute the Determinant

Identify the diagonal elements of the matrix: 5, -3, and 2. Use the formula to find the determinant:\[ \det(A) = 5 \times (-3) \times 2 \]

Step 4: Perform the Multiplication

Calculate the product:\[ \det(A) = 5 \times (-3) \times 2 = -15 \times 2 = -30 \]

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where all the elements outside the main diagonal are zero.
This makes diagonal matrices quite interesting because the calculations you perform on them are often much simpler.
In mathematical terms, a diagonal matrix \( D \) of size \( n \times n \) is represented as follows:

• The main diagonal: \( a_{ii} \), where \( i = j \)
• All other elements are zero: \( a_{ij} = 0 \), where \( i eq j \)

For example, consider the matrix \[\begin{pmatrix}5 & 0 & 0 \0 & -3 & 0 \0 & 0 & 2 \\end{pmatrix}\]In this example, the elements 5, -3, and 2 are on the main diagonal, and all other elements are zero.
This property allows us to easily calculate its determinant, as the determinant of a diagonal matrix is simply the product of its main diagonal elements.

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a square matrix.
Although it can seem complex, it's a systematic procedure that involves breaking down a larger matrix into smaller, more manageable parts.
The cofactor expansion is done along any row or column, ideally one with many zeros for simplification.
This process utilizes minors and cofactors:

• Minor: The determinant of a smaller matrix obtained by removing one row and one column from the original matrix.
• Cofactor: Derived from the minor with a sign change based on the position in the matrix, denoted by \((-1)^{i+j}\) times the minor of \(a_{ij}\).

While cofactor expansion works for any matrix, calculating the determinant of a diagonal matrix is simpler and direct by multiplying the elements of the main diagonal, as there are no non-zero elements off the diagonal.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to form a new matrix.
The process involves the dot product of rows and columns, following these steps:

• Ensure that the number of columns in the first matrix equals the number of rows in the second matrix.
• The element in the \(i^{th}\) row and \(j^{th}\) column of the resulting matrix is the sum of products.

For two matrices \( A = \begin{pmatrix} a & b \c & d \end{pmatrix} \) and \( B = \begin{pmatrix} e & f \g & h \end{pmatrix} \), the result of \( A \times B \) would be:\[\begin{pmatrix}ae + bg & af + bh \ce + dg & cf + dh \\end{pmatrix}\]Understanding matrix multiplication is essential, especially when solving larger systems with techniques like cofactor expansion.
Matrix multiplication is also part of various applications, from engineering computations to computer graphics.