Problem 3 - Algebra and Trigonometry Real Mathematics, Real People

Question

For a square matrix \(B\), the minor \(M_{23}=5 .\) What is the cofactor \(C_{23}\) of matrix \(B ?\)

Step-by-Step Solution

Answer

The cofactor C_{23} of the matrix B is -5

Step 1: Understand Matrix Positioning

Recall that the position of an element in a matrix is given as (i,j), where i is the row number and j is the column number. We note that in this problem, we're asked to calculate the cofactor for the element at position (2,3), which means i=2, j=3

Step 2: Compute Base for Exponent in Cofactor Formula

Since the base for the exponent in the cofactor calculation formula is (-1), this formula is (-1)^(i+j), we substitute the values of i and j into the formula, we get (-1)^(2+3) = (-1)^5

Step 3: Calculate Exponent Result

The result of (-1)^5 = -1

Step 4: Apply Cofactor Formula

Remembering the formula for calculating the cofactor: cofactor = (-1)^(i+j) * minor, we substitute the relevant values: C_{23} = (-1) * 5 = -5

Key Concepts

Matrix Minor

A **Matrix Minor** is an essential concept used in the study of matrices, especially when we talk about calculating determinants and cofactors. Imagine you have a square matrix. A minor is the determinant of a smaller matrix derived from the original matrix.To find a minor, you remove one specific row and one specific column from the matrix. This creates a smaller matrix, and you calculate its determinant. For instance, if you have a 3x3 matrix and you're finding the minor corresponding to the element at the second row and third column, you eliminate the second row and third column.

• Minor is specific to a single entry in the matrix.
• In our problem, the minor, denoted as \(M_{23}\), is given to be 5.
• The calculation of the minor involves only the entries of the reduced smaller matrix.

This makes understanding minors straightforward: you reduce the matrix, compute its determinant, and that's your minor.

Square Matrix

A **Square Matrix** is a type of matrix with an equal number of rows and columns, such as 2x2, 3x3, or 4x4. This balance between rows and columns is critical for many operations in linear algebra, such as finding the determinant, inverses, and cofactors.Square matrices have special properties:

• Only square matrices have a determinant.
• They are essential in solving systems of linear equations.
• Operations like eigenvalue calculation and matrix inversion apply directly to square matrices.

Since the original exercise mentions a square matrix \(B\), it means all these processes, like finding minors and cofactors, are applicable.

Cofactor Formula

**Cofactor Formula** helps us convert minors into cofactors, which are used to calculate the determinant of a matrix or its inverse. A cofactor is closely related to the minor with an added sign determined by its position.The formula to find a cofactor for an element in position \((i, j)\) is:\[C_{ij} = (-1)^{i+j} \times M_{ij}\]

• The \(i+j\) part helps determine if the multiplication with the minor results in a positive or negative sign.
• Each element has a unique cofactor based on its position.
• In the exercise, the given minor \(M_{23}=5\) results in a cofactor \(C_{23} = -5\) using the formula with \((-1)^5\), influencing the sign to remain negative.

Understanding cofactors is crucial for more complex matrix operations and solving larger systems of equations efficiently.