Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rrr} 4 & 0 & 2 \\ -3 & 2 & 1 \\ 1 & -1 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The minors and cofactors of a matrix are calculated by first obtaining the 2x2 determinant for each element of the matrix, followed by multiplication with \((-1)^{i+j}\) to get the corresponding cofactors. The resulting minor and cofactor matrices are the solution to this exercise.

Step 1: Compute Minors

Calculate the minor of each element of the matrix. Here, each element is denoted by \(a_{ij}\) where \(i\) represents rows and \(j\) represents columns. For example, for element \(a_{11} = 4\), remove the first row and first column, so the minor \(M_{11}\) is the determinant of the matrix \[ \begin{matrix} 2 & 1 \ -1 & 1 \end{matrix} \]. Carry out this process for all the elements of the 3x3 matrix.

Step 2: Determine the cofactor matrix

Having found all the minors, calculate the corresponding cofactors. The cofactor of an element \(a_{ij}\) is given by \(C_{ij} = (-1)^{i+j}M_{ij}\), where \(M_{ij}\) is the minor of \(a_{ij}\). For instance, the cofactor of \(a_{11}\) would be \(C_{11} = (-1)^{1+1}M_{11}\). Do this for each element of the 3x3 matrix.

Step 3: Compile results

The result includes the minor for every element (matrix of minors) as well as the cofactor for all elements (matrix of cofactors).

Key Concepts

Minors of a Matrix

In the context of matrix algebra, a "minor" refers to a simplified form of a determinant that's specific to one element of a larger matrix. Imagine a matrix like the one from the exercise, and let's talk about how to find a minor for, say, the element in the first row and first column, denoted as \( a_{11} \). The process is simple:

• Remove the row and column in which the element is located.
• This leaves you with a smaller matrix.
• The determinant of this smaller matrix is the "minor" of that element.

For instance, to find the minor of the element \( a_{11} = 4 \), you would write down the remaining 2x2 matrix:\[\begin{bmatrix} 2 & 1 \ -1 & 1 \end{bmatrix}\]Then, the determinant of this matrix is calculated as \( (2)(1) - (-1)(1) = 2 + 1 = 3 \), so the minor \( M_{11} = 3 \).
Remember, every element will have a corresponding minor, and you follow this process for each one.

Cofactors of a Matrix

A "cofactor" of an element in a matrix is closely related to the concept of minors, with one additional step that involves applying a sign change. Once you've found the minor for an element, obtaining the cofactor involves:

• Calculating the minor of that element.
• Multiplying the minor by \((-1)^{i+j}\), where \(i\) is the row number and \(j\) is the column number of the element.

Let's stick with our previous example of the element \(a_{11} = 4\). We already calculated its minor as \(M_{11} = 3\). To find the cofactor:
The position is the first row and first column, thus \((i+j) = 1+1 = 2\).
Compute the cofactor: \(C_{11} = (-1)^{2}\times 3 = 1 \times 3 = 3\).
The same technique applies to all matrix elements. This signed minor transformation is crucial when determining matrix determinants or performing more advanced operations like matrix inversion.

Determinants

Determinants are a fundamental component of matrix algebra, playing an essential role in systems of equations, transformations, and more. A determinant is a special number that can be calculated from the elements of a square matrix. For a 3x3 matrix, like the one in the exercise, the determinant is calculated using the formula:\[\text{det}(A) = a_{11}(C_{11}) + a_{12}(C_{12}) + a_{13}(C_{13})\]where each \(a_{ij}\) is an element of the matrix and \(C_{ij}\) is its corresponding cofactor. To compute the determinant for our example matrix:
Plug in the values for \(a_{11}\), \(a_{12}\), and \(a_{13}\) and their cofactors. This sums up to a single numerical value representing the matrix. The sign flips introduced by each cofactor ensure the determinant accurately provides insights into the matrix's properties, such as invertibility:

• If the determinant is zero, the matrix does not have an inverse, making it singular.
• A non-zero determinant means the matrix is invertible.

Understanding how to calculate determinants gives insights into the behavior and characteristics of matrices.