Problem 32 - Precalculus with Limits

Question

In Exercises 25-32, find all (a) minors and (b) cofactors of the matrix. \(\left[ \begin{array}{r} -2 & 9 & 4 \\ 7 & -6 & 0 \\ 6 & 7 & -6 \end{array} \right]\)

Step-by-Step Solution

Answer

All minors and cofactors have been calculated. Minors are: -36, -42, 25, -98, -12, -86, 24, -28, -2. Cofactors are: 36, 42, 25, 98, -12, 86, 24, 28, -2.

Step 1: Calculate minors

The minors of the matrix are calculated by taking the determinant of the 2x2 submatrix that results after eliminating the row and column of a particular element. We’ll do this for each element of the matrix. To calculate determinant of 2x2 matrix \[ startmatrix a & b \ c & d endmatrix \] we calculate as ad-bc. This yields:Minor (1, 1): `-6*(-6) - 0*7 = -36`Minor (1, 2): `7*(-6) - 0*6 = -42`Minor (1, 3): `7*7 - -6*6 = 25` Minor (2, 1): `9*(-6) - 4*7 = -98` Minor (2, 2): `-2*(-6) - 4*6 = -12` Minor (2, 3): `-2*7 - 9*6 = -86`Minor (3, 1): `9*0 - 4*(-6) = 24`Minor (3, 2): `-2*0 - 4*7 = -28`Minor (3, 3): `-2*9 - -6*7 = -2`

Step 2: Calculate cofactors

Cofactors are calculated by multiplying the minors by (-1)^(i+j), so we calculate cofactor i,j = (-1)^(i+j) * Minor i,jCofactor (1, 1): `-1^(1+1) * -36 = 36`Cofactor (1, 2): `-1^(1+2) * -42 = 42`Cofactor (1, 3): `-1^(1+3) * 25 = 25` Cofactor (2, 1): `-1^(2+1) * -98 = 98` Cofactor (2, 2): `-1^(2+2) * -12 = -12` Cofactor (2, 3): `-1^(2+3) * -86 = 86`Cofactor (3, 1): `-1^(3+1) * 24 = 24`Cofactor (3, 2): `-1^(3+2) * -28 = 28`Cofactor (3, 3): `-1^(3+3) * -2 = -2`

Key Concepts

Understanding Minors in a Matrix

In the context of matrix determinants, a minor is a critical concept that helps in breaking down a complex matrix into simpler parts. To compute a minor, you need to identify the 2x2 submatrix corresponding to any given element, say element \(a_{ij}\), in the original 3x3 matrix. You do this by eliminating the row and column that contain the element in question.

For instance, if we look at the element in the first row and first column, \(-2\) in our given matrix:

• We remove the first row and first column.
• The remaining 2x2 submatrix is composed of the elements \([-6, 0; 7, -6]\).
• To determine the minor \(M_{11}\), calculate the determinant of this submatrix: \(-6 \times -6 - 0 \times 7 = 36\).

Each element from the 3x3 matrix will have a similar minor, and repetition of this process yields all minors needed to compute the matrix's determinant. It’s easy to make mistakes, however, if one miscounts or attributes elements inaccurately.

Cofactors and Sign Adjustment

Once the minors of a matrix have been calculated, the next step involves converting them into cofactors. To obtain a cofactor, you must take the minor associated with each element of the matrix and multiply it by \((-1)^{i+j}\). Here, \((i, j)\) are the position indexes of the element whose cofactor we are calculating in the matrix.

For example:

• The cofactor for the first-row first-column element is obtained from its minor. Compute it as \((-1)^{1+1} \times M_{11} = 36\).
• The alternating sign pattern \((-1)^{i+j}\) ensures that cofactors switch in sign depending on their position in the matrix.
• In this pattern, elements in positions where the sum of the row and column indexes is even yield positive cofactors.

Cofactors play a central role in further matrix computations, including finding the determinant and in processes like matrix inversion.

Components of a 3x3 Matrix

The fundamental structure of a 3x3 matrix includes three rows and three columns, thereby containing nine elements in total. This is a basic yet powerful construct in linear algebra, often used to solve systems of linear equations, calculate determinants, and analyze vector spaces.

Understanding the arrangement:

• Every position in the matrix is identified by a unique pair of indices, such as \(a_{11}\) for the first-row first-column element.
• The upper left element, \(a_{11}\), is pivotal as it often serves as a starting point for calculations.
• Similarly, the elements \(a_{12}\), \(a_{13}\), \(a_{21}\), and so on, follow sequentially, filling up the matrix in row-major order.

Mastering the notion of a 3x3 matrix enables better handling of determinants and facilitates more complex operations like finding inverses and eigenvalues.