Problem 5 - Advanced Engineering Mathematics

Question

In Problems 5-8, suppose $$ \mathbf{A}=\left(\begin{array}{rrrr} 0 & 2 & 4 & 0 \\ 1 & 2 & -2 & 3 \\ 5 & 1 & 0 & -1 \\ 1 & 1 & 1 & 2 \end{array}\right) $$ \text { Evaluate the indicated minor determinant or cofactor. } $$ M_{33} $$

Step-by-Step Solution

Answer

The value of the minor determinant \(M_{33}\) is 2.

Step 1: Identify the Minor

The minor \(M_{33}\) corresponds to the determinant of the 3x3 matrix obtained by deleting the third row and the third column of matrix \(\mathbf{A}\).

Step 2: Form the Submatrix

By removing the third row and third column of \(\mathbf{A}\), we get the 3x3 submatrix: \[\mathbf{B}=\begin{pmatrix}0 & 2 & 0 \ 1 & 2 & 3 \ 1 & 1 & 2\end{pmatrix}.\]

Step 3: Calculate the Determinant of the Submatrix

Use the formula for the determinant of a 3x3 matrix: \[\text{det}(\mathbf{B}) = a(ei-fh) - b(di-fg) + c(dh-eg).\] For matrix \(\mathbf{B}\), substitute the values accordingly: \[\text{det}(\mathbf{B}) = 0(2 \times 2 - 3 \times 1) - 2(1 \times 2 - 3 \times 1) + 0(1 \times 1 - 2 \times 1).\] Simplifying, we find \[\text{det}(\mathbf{B}) = 0 - 2(2 - 3) + 0 = 2.\]

Step 4: Conclusion

The value of the minor determinant \(M_{33}\) is 2.

Key Concepts

Matrix Theory

Matrix Theory is the branch of mathematics that explores arrays of numbers and their properties. Matrices are fundamental in various scientific fields, especially when solving linear equations and transforming geometric data. In simple terms, a matrix is a rectangular array of numbers arranged in rows and columns.

• Each number inside a matrix is called an element.
• The size or dimension of a matrix is given by the number of its rows and columns.

To solve certain problems involving matrices, we often deal with minor determinants and cofactors. A minor is the determinant of a smaller matrix that we get by omitting one row and one column from a larger matrix. These miniscule-sized matrices become the building blocks for more complex operations like finding the determinant of the bigger matrix itself.

Cofactor Expansion

Cofactor Expansion is a crucial method in linear algebra used to calculate the determinant of a matrix. It simplifies a larger determinant into more manageable calculations using minors and cofactors.
A cofactor, in essence, is a minor determinant with a sign assigned based on the position of the element in the matrix. The sign is calculated using \[(-1)^{i+j}\]where \(i\) is the row number and \(j\) is the column number of the element:

• If both \(i\) and \(j\) are even or both are odd, the cofactor is positive.
• If one of \(i\) or \(j\) is even and the other is odd, the cofactor is negative.

To use cofactor expansion, you select a row or column, and for each element in that row or column, you multiply the element by its corresponding cofactor. Finally, sum the results to obtain the determinant of the matrix.

3x3 Matrix Determinant

The determinant of a 3x3 matrix is a specific case where the general method of determinant calculation becomes intuitive and straightforward. For a 3x3 matrix:\[\mathbf{B} = \begin{pmatrix}a & b & c \ d & e & f \ g & h & i\end{pmatrix},\]The determinant can be calculated using the following formula:\[\text{det}(\mathbf{B}) = a(ei - fh) - b(di - fg) + c(dh - eg).\]This method involves a systematic approach where:

• Each element of the first row (\(a, b, c\)) is multiplied by the determinant of the 2x2 matrix left behind when the row and column of the element are removed.
• Subtract or add this product depending on the position's cofactor sign.

Knowing how to manage these signs and operations will give you the determinant of the matrix, representing different properties like volume when dealing with vectors, and ensuring matrices are invertible.