Problem 12 - Precalculus: Mathematical for Calculus

Question

Evaluate the minor and cofactor using the matrix \(A\) $$A=\left[\begin{array}{rrr} 1 & 0 & \frac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{array}\right]$$ $$M_{13}, A_{13}$$

Step-by-Step Solution

Answer

The minor \(M_{13}\) is 0, and the cofactor \(A_{13}\) is 0.

Step 1: Identify the Submatrix

To find the minor \(M_{13}\), we need the submatrix obtained by removing the first row and third column from matrix \(A\). This leaves us with the matrix: \[ \begin{bmatrix} -3 & 5 \ 0 & 0 \end{bmatrix} \]

Step 2: Calculate the Minor \(M_{13}\)

The minor \(M_{13}\) is the determinant of the submatrix \( \begin{bmatrix} -3 & 5 \ 0 & 0 \end{bmatrix} \). The determinant is calculated as \((-3)(0) - (5)(0) = 0\). Thus, \(M_{13} = 0\).

Step 3: Calculate the Cofactor \(A_{13}\)

The cofactor \(A_{13}\) is given by \( (-1)^{1+3} \times M_{13}\). Since \( (-1)^{1+3} = (-1)^4 = 1 \), \(A_{13} = 1 \times 0 = 0\).

Key Concepts

Determinant

In the world of matrices, understanding determinants is essential for working with matrix calculations. A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, including whether the matrix is invertible or singular.
Determinants are particularly useful in solving systems of linear equations, calculus, and finding eigenvalues.

• If the determinant of a matrix is zero, the matrix is singular and does not have an inverse.
• If the determinant is non-zero, the matrix is invertible.

Calculating the determinant might seem tricky, but it's usually done through understanding the pattern of multiplications and subtractions of elements. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \).
For larger matrices, the process involves expanding along a row or column, multiplying each element by its corresponding cofactor, and summing the results. This technique involves calculating minors, which we will further explain in the next section.

Minor

The concept of a minor in a matrix is critical when calculating determinants of matrices larger than 2x2. A minor of a specific element in a matrix is the determinant of what's left after removing the row and column in which this particular element appears.
This simplification helps in computing the determinant of larger matrices by breaking it down into smaller, more manageable parts.
Let's take the matrix \( A = \begin{bmatrix} 1 & 0 & \frac{1}{2} \ -3 & 5 & 2 \ 0 & 0 & 4 \end{bmatrix} \) as an example, especially focusing on the element in the first row and third column (1,3 position). The minor \( M_{13} \) is calculated by removing the first row and third column, resulting in the 2x2 submatrix \( \begin{bmatrix} -3 & 5 \ 0 & 0 \end{bmatrix} \).
The determinant of this submatrix, given its structure (since it involves a row of zeros), is straightforward: \(-3 imes 0 - 5 imes 0 = 0\). As a result, the minor \( M_{13} \) is 0.

Submatrix Removal

Submatrix removal is a key step when working with minors and cofactors. This process of matrix reduction involves cutting down a larger matrix into a smaller one by removing specific rows and columns, usually to ease the computation of determinants or minors.
To explain further, let's revisit our matrix \( A = \begin{bmatrix} 1 & 0 & \frac{1}{2} \ -3 & 5 & 2 \ 0 & 0 & 4 \end{bmatrix} \). If we're tasked with finding the minor \( M_{13} \), we need to remove the first row and the third column. This operation leaves us with the submatrix \( \begin{bmatrix} -3 & 5 \ 0 & 0 \end{bmatrix} \).
This reduced matrix is then used to find the determinant, a critical step in both the minor and cofactor calculations. The process of submatrix removal simplifies complex matrices, allowing for easier computation and analysis. This technique emphasizes the importance of accuracy in identifying which rows and columns to remove, as it directly affects subsequent calculations.