Problem 11 - Precalculus: Mathematical for Calculus

Question

Evaluate the minor and cofactor using the matrix \(A\) $$A=\left[\begin{array}{rrr} 1 & 0 & \frac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{array}\right]$$ $$M_{12}, A_{12}$$

Step-by-Step Solution

Answer

The minor \(M_{12}\) is -12 and the cofactor \(A_{12}\) is 12.

Step 1: Identify the Minor

The minor, denoted as \(M_{12}\), is the determinant of the matrix that remains after removing the first row and the second column of matrix \(A\). The remaining 2x2 matrix is:\[\begin{bmatrix}-3 & 2 \ 0 & 4\end{bmatrix}\].

Step 2: Calculate the Determinant

To find \(M_{12}\), calculate the determinant of the 2x2 matrix from Step 1:\[M_{12} = (-3)(4) - (2)(0) = -12\]. Thus, \(M_{12} = -12\).

Step 3: Identify the Sign for the Cofactor

The cofactor \(A_{ij}\) is given by \((-1)^{i+j}M_{ij}\). For \(A_{12}\), here \(i=1\) and \(j=2\), hence the sign is \((-1)^{1+2} = (-1)^{3} = -1\).

Step 4: Calculate the Cofactor

Multiply the minor \(M_{12}\) by the sign identified in Step 3:\[A_{12} = (-1)(-12) = 12\].

Key Concepts

3x3 Matrices

When we talk about 3x3 matrices, we're dealing with a matrix that consists of three rows and three columns. Each element in the matrix is accessed by its row and column position. For example, in the matrix \[A = \begin{bmatrix}1 & 0 & \frac{1}{2} \-3 & 5 & 2 \0 & 0 & 4\end{bmatrix}\]the element in the first row and second column is 0.
A 3x3 matrix is common in various calculations, including transformations and operations in physics and computer graphics. It can represent a variety of real-world systems or data structures.
Understanding how to manipulate matrices, such as calculating minors and cofactors, helps in solving systems of equations and finding solutions efficiently. It is the foundational step before diving into determinants and matrix algebra.

Determinants

Determinants are a powerful tool in matrix algebra. For a 3x3 matrix, the determinant is a special scalar value that provides essential information about the matrix, such as whether it is invertible. The formula for the determinant of a 3x3 matrix \[A = \begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\]is:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Understanding how to find and interpret this value can unravel many characteristics of the matrix.
It assists in computational efficiency when calculating eigenvalues or solving linear equations. In the exercise given, calculating a minor, such as \(M_{12}\), involves finding the determinant of the 2x2 matrix after omitting specific rows and columns. This practice also reveals the interrelation between individual components of the matrix, contributing to the overall determinant calculation.

Matrix Algebra

Matrix algebra involves various operations, including addition, subtraction, multiplication, and finding inverses of matrices. These operations are crucial in solving mathematical equations involving matrices.
A significant part of matrix algebra is understanding minors and cofactors. A minor of an element in a matrix is the determinant of the submatrix formed by deleting its row and column. A cofactor, on the other hand, takes into account a sign factor determined by its position.

• A minor is calculated by removing elements from the matrix.
• A cofactor adjusts the minor by applying an alternating plus or minus sign.

Knowing these concepts allows you to compute the determinant of larger matrices using expansion by minors, facilitating the inverse calculation and other algebraic manipulations. This concept is at the heart of many real-world applications, ranging from computer graphics to systems engineering, also aiding in understanding more complex topics like eigenvectors and eigenvalues.