Problem 13 - Algebra and Trigonometry

Question

Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{11}, A_{11} $$

Step-by-Step Solution

Answer

The minor \( M_{11} \) is 20, and the cofactor \( A_{11} \) is also 20.

Step 1: Identify the Element

Find the element from matrix \(A\) located at the first row, first column, which is \(a_{11} = 1\).

Step 2: Determine the Minor

To find the minor \( M_{11} \), remove the first row and first column from the matrix \( A \) and evaluate the determinant of the resulting \(2 \times 2\) matrix:\[\left| \begin{array}{cc}5 & 2 \0 & 4 \\end{array} \right|\]The determinant is calculated as:\( (5)(4) - (0)(2) = 20 \). Thus, \( M_{11} = 20 \).

Step 3: Calculate the Cofactor

The cofactor \( A_{11} \) is given by the formula \( A_{ij} = (-1)^{i+j}M_{ij} \). For \( A_{11} \), we have:\[ A_{11} = (-1)^{1+1}M_{11} = (1)(20) = 20 \]Thus, \( A_{11} = 20 \).

Key Concepts

Minor of a Matrix

Understanding the concept of a minor is crucial in the study of matrix algebra. The minor of a matrix, often denoted as \( M_{ij} \), refers to the determinant of a smaller matrix that is created by eliminating one row and one column from the original matrix. The specific row and column to be removed correspond to the position of the element \( a_{ij} \) whose minor is being calculated. This smaller matrix is referred to as a submatrix.

For example, to find the minor \( M_{11} \) of a 3x3 matrix \( A \), you would remove the first row and the first column from the original matrix. This leaves you with a 2x2 matrix, from which you calculate the determinant. This determinant value is the minor.

Minors are essential as they form the building blocks for finding determinants of larger matrices, making them an important concept for anyone studying linear algebra.

Cofactor of a Matrix

A cofactor is another fundamental concept in matrix algebra, directly related to the minor of the matrix. The cofactor, denoted as \( A_{ij} \), is derived from the minor \( M_{ij} \) by applying a sign change based on the position of the element \( a_{ij} \) within the matrix. The formula used is:

• \( A_{ij} = (-1)^{i+j}M_{ij} \)

This formula helps in determining the sign of the cofactor. The rule \((-1)^{i+j}\) means that the sign of the cofactor alternates depending on whether the sum of the row index \( i \) and the column index \( j \) is even or odd.

In practical terms, the cofactor takes the minor of an element, \( M_{ij} \), and adjusts the sign according to its position. This is an important step when calculating the determinant of a larger square matrix and utilizing matrix algebra tools like cofactor expansion.

Determinant of a 2x2 Matrix

Calculating the determinant of a 2x2 matrix is a straightforward process that forms the basis of understanding determinants in matrix algebra. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant, denoted as \( \det \), is computed using the formula:

• \( \det = ad - bc \)

The determinant is a single value that offers information about the matrix, such as whether it can be inverted. In the context of a 3x3 matrix like matrix \( A \), the determinant of the 2x2 submatrix is used when calculating the minor or cofactor.

The straightforward nature of computing a 2x2 determinant makes it a foundational skill. It's a stepping stone to evaluating determinants of higher-dimensional matrices, making it crucial to master early on in matrix algebra.