Cofactor Calculation: Matrix H Example

In linear algebra, understanding matrices and their properties is super important. One of these properties is the cofactor, which is a key part of finding the inverse of a matrix and calculating determinants. Today, we're going to dive into how to calculate a specific cofactor from a given matrix. So, let's break it down and make it easy to understand!

What are Cofactors?

Before we jump into calculating cofactors, let's quickly recap what they are. Cofactors come from something called a minor. Imagine you have a matrix, and you want to find the minor of a specific element. What you do is you cross out the row and column that element is in, and then you find the determinant of the smaller matrix that's left. That determinant is the minor.

Now, the cofactor is just a minor with a sign adjustment. The formula to find the cofactor ${C_{ij}}$ of an element in the ${i}$-th row and ${j}$-th column is:

${C_{ij} = (-1)^{i+j} M_{ij}}$

Where ${M_{ij}}$ is the minor of the element in the ${i}$-th row and ${j}$-th column. The ${(-1)^{i+j}}$ part just flips the sign depending on whether ${i+j}$ is even or odd. If ${i+j}$ is even, the cofactor is the same as the minor. If ${i+j}$ is odd, the cofactor is the negative of the minor.

So, in simple terms:

• Find the minor by calculating the determinant of the smaller matrix after removing the row and column of your element.
• Determine the sign based on the position of the element using ${(-1)^{i+j}}$.
• Multiply the minor by the sign to get the cofactor.

Problem

We are given the matrix:

${H = \begin{pmatrix} 5 & -2 & 3 \\ 1 & 4 & 0 \\ 2 & 3 & 1 \end{pmatrix}}$

Our task is to find the cofactor ${C_{13}}$.

Step-by-Step Solution to Find ${C_{13}}$

To find the cofactor ${C_{13}}$ of the matrix ${H}$, we need to follow these steps carefully:

Step 1: Identify the Element

First, let's identify the element for which we need to find the cofactor. The notation ${C_{13}}$ means we are looking for the cofactor of the element in the first row and third column of matrix ${H}$. Looking at the matrix:

${H = \begin{pmatrix} 5 & -2 & \textbf{3} \\ 1 & 4 & 0 \\ 2 & 3 & 1 \end{pmatrix}}$

The element in the first row and third column is 3. So, we want to find the cofactor associated with this element.

Step 2: Find the Minor ${M_{13}}$

To find the minor ${M_{13}}$, we need to remove the first row and the third column from matrix ${H}$. This leaves us with a 2x2 matrix:

${\begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix}}$

Now, we calculate the determinant of this 2x2 matrix. The determinant of a 2x2 matrix ${\begin{pmatrix} a & b \\ c & d \end{pmatrix}}$ is given by ${ad - bc}$. So, for our matrix, the determinant is:

${M_{13} = (1 \times 3) - (4 \times 2) = 3 - 8 = -5}$

Thus, the minor ${M_{13}}$ is -5.

Step 3: Determine the Sign

Next, we need to determine the sign adjustment for the cofactor. The sign is determined by ${(-1)^{i+j}}$, where ${i}$ is the row number and ${j}$ is the column number. In our case, we are finding ${C_{13}}$, so ${i = 1}$ and ${j = 3}$. Thus, we have:

${(-1)^{1+3} = (-1)^4 = 1}$

Since ${(-1)^4 = 1}$, the sign is positive.

Step 4: Calculate the Cofactor ${C_{13}}$

Now, we can calculate the cofactor ${C_{13}}$ using the formula:

${C_{13} = (-1)^{1+3} M_{13}}$

We found that ${M_{13} = -5}$ and ${(-1)^{1+3} = 1}$, so:

${C_{13} = 1 \times (-5) = -5}$

Therefore, the cofactor ${C_{13}}$ is -5.

Final Answer

The entry of the cofactor ${C_{13}}$ for the given matrix ${H}$ is:

A. -5

Key Concepts Used

To solve this problem, we used the following concepts:

• Matrix: A rectangular array of numbers, symbols, or expressions arranged in rows and columns.
• Determinant: A scalar value that can be computed from the elements of a square matrix.
• Minor: The determinant of the submatrix formed by deleting the ${i}$-th row and ${j}$-th column of the original matrix.
• Cofactor: The minor with a sign adjustment, calculated as ${C_{ij} = (-1)^{i+j} M_{ij}}$.

Additional Tips

• Always double-check your calculations, especially when computing determinants, as a small error can lead to an incorrect answer.
• Remember the sign convention for cofactors: the sign alternates as you move across the matrix, starting with a positive sign in the top-left corner.
• Practice with different matrices to become comfortable with the process of finding minors and cofactors.

Why are Cofactors Important?

Cofactors are important in matrix algebra for several reasons. First off, they're essential for finding the inverse of a matrix. You know, that matrix that, when you multiply it by the original, gives you the identity matrix? Yeah, to find that, you need cofactors. Specifically, you use cofactors to create the adjugate (or adjoint) matrix, which is just the transpose of the cofactor matrix. Then, you divide each element of the adjugate matrix by the determinant of the original matrix to get the inverse.

So, in short, cofactors are a critical component in calculating the inverse of a matrix. The inverse is used to solve systems of linear equations, which pop up all over the place in math, science, engineering, and even economics.

Practice Problems

To solidify your understanding, here are a few practice problems:

1. Given matrix ${A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}}$, find ${C_{11}}$ and ${C_{22}}$.
2. Given matrix ${B = \begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 0 & 2 & -2 \end{pmatrix}}$, find ${C_{12}}$.

Conclusion

Calculating cofactors might seem a bit complex at first, but once you understand the steps and practice a bit, it becomes straightforward. Remember to find the minor, determine the sign, and then calculate the cofactor. With these skills, you'll be well-equipped to tackle more advanced matrix problems. Keep practicing, and you'll master these concepts in no time! Also, don't be afraid to ask questions and seek help when needed. Happy calculating, folks!