Problem 28 - Precalculus with Limits

Question

In Exercises 25-32, find all (a) minors and (b) cofactors of the matrix. \(\left[ \begin{array}{r} -6 & 5 \\ 7 & -2 \end{array} \right]\)

Step-by-Step Solution

Answer

The minors of the matrix are -2, 7, 5, and -6 while the cofactors are -2, -7, -5, and -6 respectively.

Step 1: Find the Minors

To find the minors, for each element in the matrix, omit the corresponding row and column and compute the determinant of the leftover element. In a 2x2 matrix, the minor of an element, for example, the 1st row, 1st column (-6), is simply the remaining element, which is -2. Similarly, the minor of the element in the 1st row, 2nd column (5) is 7. In a similar fashion, the minor of 7 is 5 and the minor of -2 is -6.

Step 2: Find the Cofactors

Cofactors are calculated similarly as minors, but we also take into account the position of the element in the matrix (i,j), by using the formula (-1)^(i+j). For our matrix the cofactor of -6 is \((-1)^{1+1} \times -2 = -2\), the cofactor of 5 is \((-1)^{1+2} \times 7 = -7\), the cofactor of 7 is \((-1)^{2+1} \times 5 = -5\), and the cofactor of -2 is \((-1)^{2+2} \times -6 = -6\). Find the other cofactors in a similar way.

Step 3: List the Minors and Cofactors

Listing down all the minors and cofactors of the elements of the matrix, we have: Minors of -6 is -2, 5 is 7, 7 is 5 and -2 is -6. Cofactors of -6 is -2, 5 is -7, 7 is -5 and -2 is -6 .

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It is extremely useful in various calculations in matrix algebra, including finding the inverse of a matrix, and understanding systems of linear equations. For a 2x2 matrix, the formula to compute the determinant is straightforward and simple. If the matrix is represented as \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is given by the formula \(ad - bc\).

When you calculate the determinant of a 2x2 matrix, you essentially find the difference between the product of its diagonal elements. This makes the process quick and efficient. The determinant provides insights into the matrix's properties, like if the matrix is invertible (a non-zero determinant) or singular (a zero determinant). Understanding determinants helps students grasp more complex topics in linear algebra.

2x2 Matrix

A 2x2 matrix is the simplest form of a square matrix, consisting of two rows and two columns. Its compact size makes it particularly friendly for performing basic calculations, including determining minors and cofactors.

• Each element in a 2x2 matrix can be labeled based on its position, specifically by the row and column it resides in.
• Since it only has four elements, finding minors and cofactors involves simple arithmetic operations.

For instance, consider the matrix \(\left[ \begin{array}{r} -6 & 5 \ 7 & -2 \end{array} \right]\). Here each element, such as -6, is positioned in the first row and first column. Understanding its position is important for calculating the minors and cofactors accurately.

The 2x2 matrix is often the starting point for students to familiarize themselves with concepts in matrix algebra, serving as a foundation before moving on to larger matrices.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with operations on matrices. Matrices are arrays of numbers organized in rows and columns, useful in numerous fields such as physics, computer science, and engineering. Basic operations in matrix algebra include addition, subtraction, multiplication, and finding determinant, minors, and cofactors.

Being able to compute determinants and cofactors is an essential skill in matrix algebra.

• Calculating affects how we solve linear equations, as it determines whether a system has a unique solution.
• Minors help to find the determinant of larger matrices and are used when computing Eigenvalues and Eigenvectors.
• Cofactors corroborate the expansion of determinants and play a critical role in finding matrix inverses.

Mastering these operations helps students solve complex problems involving transformations, solve systems of equations, and model real-world situations. With practice, handling the manipulation of matrices becomes more intuitive and accessible.