Problem 3 - Advanced Engineering Mathematics

Question

In Problems \(1-4\), suppose $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 4 \\ 1 & -1 & 2 \\ -2 & 3 & 5 \end{array}\right) $$ Evaluate the indicated minor determinant or cofactor. $$ C_{13} $$

Step-by-Step Solution

Answer

The cofactor \( C_{13} \) is 1.

Step 1: Identify the Cofactor

To find the cofactor \( C_{13} \), we first need to identify the element in the 1st row and 3rd column of matrix \( \mathbf{A} \). The element in position \( a_{13} \) is \( 4 \).

Step 2: Form the Submatrix

Remove the 1st row and the 3rd column from matrix \( \mathbf{A} \) to form a submatrix. The submatrix that corresponds to \( a_{13} \) is: \[ \begin{pmatrix} 1 & -1 \ -2 & 3 \end{pmatrix} \]

Step 3: Find the Determinant of Submatrix

Calculate the determinant of the 2x2 submatrix:\[ \text{det} \begin{pmatrix} 1 & -1 \ -2 & 3 \end{pmatrix} = (1)(3) - (-1)(-2) = 3 - 2 = 1 \]

Step 4: Apply Cofactor Sign

The cofactor \( C_{13} \) includes a sign based on its position. For the position (1,3), use the sign \((-1)^{1+3} = (-1)^4 = 1\). Thus, the cofactor is calculated as:\[ C_{13} = 1 \times 1 = 1 \]

Key Concepts

Understanding Cofactors

Cofactors play a significant role in matrix algebra and the computation of determinants. They are particularly important because they help break down complex matrices into simpler ones. When dealing with a matrix, each element has an associated cofactor. This is especially crucial when calculating the determinant of larger matrices.
To find a cofactor, you must look at a specific element within the matrix. The cofactor is computed by focusing on this element and its unique position in the matrix. The cofactor for an element located in the first row and third column, like in our example, is not just the determinant of the submatrix. It also includes a sign factor which is influenced by the element's position. A positive or negative sign is applied, commonly calculated using \[(-1)^{i+j}\]. For position \((1, 3)\), this results in \((-1)^4\) which is positive, hence the cofactor calculation factor is forgotten when multiplying the submatrix's determinant.

Submatrix Formation

Forming a submatrix is an essential step when dealing with determinants and cofactors. The process requires removing specific rows and columns from the original matrix. This smaller matrix is known as a submatrix.
To form a submatrix for a particular element, you need to:

• Eliminate the row where the element resides.
• Remove the column of that element as well.

For example, in our exercise, to form the submatrix related to the element \(a_{13}\), we remove the first row and the third column. This leaves us with a manageable \(2x2\) matrix.This submatrix then becomes the base for determinant calculation in the following steps.

Determinant Calculation

Determinants are fundamental when it comes to matrices. They provide vital information such as whether a matrix is invertible. For a \(2x2\) submatrix, the determinant is straightforward to calculate and follows the formula:\[\text{det} \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc\]Using our example with elements \(1, -1, -2,\) and \(3\), the determinant of the submatrix becomes: \((1)(3) - (-1)(-2) = 3 - 2 = 1\).This determinant is then used in computing the cofactor. It's important to note that for larger matrices, the determinant calculation involves breaking the matrix down into simpler submatrices, making the step-by-step process crucial for easier computation.