Problem 1 - 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. $$ M_{12} $$

Step-by-Step Solution

Answer

The minor \(M_{12}\) is 9.

Step 1: Identify the Minor

The minor, denoted as \(M_{12}\), corresponds to the determinant of the matrix obtained by deleting the first row and the second column of matrix \(\mathbf{A}\).

Step 2: Form the Submatrix

Remove the first row and second column from matrix \(\mathbf{A}\). The resulting submatrix is: \[ l \begin{bmatrix} 1 & 2 \ -2 & 5 \end{bmatrix} \]

Step 3: Calculate the Determinant of the Submatrix

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \(ad - bc\). Applying this formula: \[ \det = (1)(5) - (2)(-2) = 5 - (-4) = 5 + 4 = 9 \] Thus, the determinant of the submatrix is 9.

Key Concepts

Matrix Algebra

Matrix algebra is a crucial part of linear algebra and involves operations with matrices. A matrix is essentially an array of numbers arranged in rows and columns. In matrix algebra, you can perform operations such as addition, subtraction, multiplication, and finding determinants. These operations are fundamental in solving various types of problems in mathematics and engineering.
- **Addition and Subtraction** involve element-wise operations: this means adding or subtracting corresponding elements of matrices of the same dimension.
- **Matrix Multiplication** is a bit more complex, requiring the number of columns in the first matrix to equal the number of rows in the second. The resulting matrix has dimensions of rows from the first and columns from the second matrix.
- The **Determinant** of a matrix provides information about the matrix, such as whether it has an inverse. It's a scalar value computed from the elements of a square matrix. For a 2x2 matrix, the determinant is calculated using the formula: \( ad - bc \). This concept extends to larger matrices using more complex methods like cofactor expansions.

Cofactors

Cofactors are integral in calculating the determinant of a larger square matrix and understanding the matrix's properties. A cofactor is essentially the minor of a matrix element, multiplied by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column numbers of that element.
- To get the **Cofactor of an Element**, you must first calculate the minor of that element by removing its row and column from the matrix and then find the determinant of this submatrix.
- Multiply this determinant by \((-1)^{i+j}\) to get the cofactor.
- The significance of cofactors becomes evident when using the cofactor expansion formula to calculate the determinant of larger matrices. This involves summing the products of each element of a row or column and its corresponding cofactor.

Minor of a Matrix

The minor of a matrix is a fundamental concept for understanding determinants and matrix inversions. It refers to the determinant of a certain smaller square matrix formed from a bigger one by removing a row and a column.
- **Calculate a Minor for an element in a matrix**, first identify the location of the element by its row and column.
- Remove the specific row and column containing the element of interest.
- The resulting smaller matrix is termed a submatrix. The determinant of this submatrix is the minor of the element.
- **Example:** In the exercise, the minor \(M_{12}\) involves taking out the 1st row and 2nd column of the matrix \(\mathbf{A}\), resulting in a submatrix whose determinant is then calculated. For \(\mathbf{A}\), this is a 2x2 matrix, and the determinant we computed is \(9\), thus making it the minor for the specified position.