Problem 14 - Algebra and Trigonometry

Question

\(9-14\) Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{32}, A_{32} $$

Step-by-Step Solution

Answer

The minor is 4 and the cofactor is -4.

Step 1: Determine Minor M_32

To find minor \( M_{32} \), remove the 3rd row and 2nd column of the matrix. The remaining 2x2 matrix is \( \left[ \begin{array}{cc} 1 & \frac{1}{2} \ 0 & 4 \end{array} \right] \). The minor is the determinant of this matrix: \( M_{32} = \left(1\cdot4\right) - \left(\frac{1}{2}\cdot0\right) = 4 - 0 = 4 \)

Step 2: Determine Cofactor A_32

The cofactor \( A_{32} \) is given by the formula \( A_{32} = (-1)^{3+2} \cdot M_{32} \). Here, \((-1)^{3+2} = (-1)^5 = -1\). Thus, \( A_{32} = -1 \cdot 4 = -4 \).

Key Concepts

Matrix Determinant

The concept of a determinant is fundamental in working with matrices. The determinant of a matrix is a scalar value that provides important properties about the matrix. In practical terms, it can indicate whether a matrix is invertible or not. For 2x2 matrices, such as the minor matrix found in the problem, the determinant can be calculated using the formula: \(\text{det}(\begin{bmatrix} a & b \ c & d \end{bmatrix}) = ad - bc\).

Let's break it down:

• Choose the elements of the matrix: here, we have the matrix \(\begin{bmatrix} 1 & \frac{1}{2} \ 0 & 4 \end{bmatrix}\).
• The formula becomes \(1\cdot4 - \frac{1}{2}\cdot0 = 4\).
• Thus, in this example, the determinant of the 2x2 matrix is \(4\).

Understanding determinants is crucial because it sets the foundation for determining minors and cofactors of a matrix. Remember that the determinant's calculation differs based on the matrix's size and structure.

Matrix Minor

A matrix minor is essential for higher-dimensional determinants. The minor of an element in a matrix is the determinant of a smaller matrix, derived by removing the row and column containing that element. This reduced matrix, particularly in this exercise, leads to a 2x2 matrix. Here's how you find it:

• Focus on the desired element—in the exercise, the element is positioned at \((3,2)\).
• Remove the 3rd row and 2nd column from the original 3x3 matrix \(A\).
• The remaining elements form the matrix \(\begin{bmatrix} 1 & \frac{1}{2} \ 0 & 4 \end{bmatrix}\).

The determinant of this new matrix is the minor, \(M_{32}=4\). Calculating minors is a critical part of finding cofactors, which, in turn, are used in further operations like computing the determinant of larger matrices.

Matrix Cofactor

Cofactors build on the concept of minors to assist in calculations needed for matrix operations such as finding the determinant of larger matrices or solving systems of linear equations. A cofactor is essentially a signed version of the minor. The sign is determined by the position of the element in the matrix: if the sum of the row and column is even, the sign is positive; if odd, the sign is negative. Here's how you calculate it:

• Use the minor already found: \(M_{32} = 4\).
• Identify the position of the element in the original matrix, here \((3,2)\).
• Calculate the sign using \((-1)^{3+2} = -1\).
• The cofactor is then \(A_{32} = -1 \cdot 4 = -4\).

In this example, the cofactor \(A_{32}\) is \(-4\). Cofactors are crucial in operations such as finding the inverse of a matrix, as they help construct the adjugate, a crucial component in these calculations.