Matrix Cofactors: A Step-by-Step Guide

Are you diving into the world of linear algebra and encountering those mysterious "cofactors"? Don't worry, you're not alone! Calculating cofactors for a matrix might seem a bit daunting at first, but with a clear understanding of the process, it becomes quite manageable. Let's break down how to find the cofactors for each row of a given matrix, using $A=\left[\begin{array}{ccc} 12 & 23 & -6 \\ 11 & -8 & 13 \\ 7 & 21 & 15 \end{array}\right]$ as our example.

Understanding Cofactors

Before we get our hands dirty with calculations, let's quickly recap what a cofactor is. In essence, a cofactor is a specific number associated with each element in a matrix. It's a crucial component in several matrix operations, most notably in finding the determinant and the inverse of a matrix. Each cofactor, denoted as $C_{ij}$, is calculated by multiplying the minor of the element at position $(i, j)$ by $(-1)^{i+j}$. The minor, in turn, is the determinant of the submatrix formed by removing the $i$-th row and $j$-th column from the original matrix.

So, to find a cofactor $C_{ij}$, we first need to identify the element $a_{ij}$ in the matrix. Then, we eliminate the row and column that element belongs to, creating a smaller matrix. We calculate the determinant of this smaller matrix – that's our minor. Finally, we multiply this minor by either 1 or -1, depending on the sum of the row and column indices ($i+j$). If $i+j$ is even, we multiply by 1 (so the cofactor equals the minor). If $i+j$ is odd, we multiply by -1 (so the cofactor is the negative of the minor).

Calculating Cofactors for the First Row

Let's start with the first row of matrix $A$. Our matrix is $A=\left[\begin{array}{ccc} 12 & 23 & -6 \\ 11 & -8 & 13 \\ 7 & 21 & 15 \end{array}\right]$. The elements in the first row are $a_{11}=12$, $a_{12}=23$, and $a_{13}=-6$. We need to calculate the cofactors $C_{11}$, $C_{12}$, and $C_{13}$.

• For $a_{11} = 12$ (position (1,1)): The minor $M_{11}$ is the determinant of the submatrix formed by removing the first row and first column: $\left|\begin{array}{cc} -8 & 13 \\ 21 & 15 \end{array}\right|$. $M_{11} = (-8 \times 15) - (13 \times 21) = -120 - 273 = -393$. The cofactor $C_{11} = (-1)^{1+1} \times M_{11} = (1) \times (-393) = \textbf{-393}$.

• For $a_{12} = 23$ (position (1,2)): The minor $M_{12}$ is the determinant of the submatrix formed by removing the first row and second column: $\left|\begin{array}{cc} 11 & 13 \\ 7 & 15 \end{array}\right|$. $M_{12} = (11 \times 15) - (13 \times 7) = 165 - 91 = 74$. The cofactor $C_{12} = (-1)^{1+2} \times M_{12} = (-1) \times 74 = \textbf{-74}$.

• For $a_{13} = -6$ (position (1,3)): The minor $M_{13}$ is the determinant of the submatrix formed by removing the first row and third column: $\left|\begin{array}{cc} 11 & -8 \\ 7 & 21 \end{array}\right|$. $M_{13} = (11 \times 21) - (-8 \times 7) = 231 - (-56) = 231 + 56 = 287$. The cofactor $C_{13} = (-1)^{1+3} \times M_{13} = (1) \times 287 = \textbf{287}$.

So, the cofactors of the first-row entries are -393, -74, and 287.

Calculating Cofactors for the Second Row

Now, let's move on to the second row of matrix $A$. The elements here are $a_{21}=11$, $a_{22}=-8$, and $a_{23}=13$. We need to find $C_{21}$, $C_{22}$, and $C_{23}$. Remember, the sign changes based on the sum of the row and column indices.

• For $a_{21} = 11$ (position (2,1)): The minor $M_{21}$ is the determinant of the submatrix formed by removing the second row and first column: $\left|\begin{array}{cc} 23 & -6 \\ 21 & 15 \end{array}\right|$. $M_{21} = (23 \times 15) - (-6 \times 21) = 345 - (-126) = 345 + 126 = 471$. The cofactor $C_{21} = (-1)^{2+1} \times M_{21} = (-1) \times 471 = \textbf{-471}$.

• For $a_{22} = -8$ (position (2,2)): The minor $M_{22}$ is the determinant of the submatrix formed by removing the second row and second column: $\left|\begin{array}{cc} 12 & -6 \\ 7 & 15 \end{array}\right|$. $M_{22} = (12 \times 15) - (-6 \times 7) = 180 - (-42) = 180 + 42 = 222$. The cofactor $C_{22} = (-1)^{2+2} \times M_{22} = (1) \times 222 = \textbf{222}$.

• For $a_{23} = 13$ (position (2,3)): The minor $M_{23}$ is the determinant of the submatrix formed by removing the second row and third column: $\left|\begin{array}{cc} 12 & 23 \\ 7 & 21 \end{array}\right|$. $M_{23} = (12 \times 21) - (23 \times 7) = 252 - 161 = 91$. The cofactor $C_{23} = (-1)^{2+3} \times M_{23} = (-1) \times 91 = \textbf{-91}$.

Therefore, the cofactors of the second-row entries are -471, 222, and -91.

Calculating Cofactors for the Third Row

Finally, let's tackle the third row of matrix $A$. The elements are $a_{31}=7$, $a_{32}=21$, and $a_{33}=15$. We need to compute $C_{31}$, $C_{32}$, and $C_{33}$.

• For $a_{31} = 7$ (position (3,1)): The minor $M_{31}$ is the determinant of the submatrix formed by removing the third row and first column: $\left|\begin{array}{cc} 23 & -6 \\ -8 & 13 \end{array}\right|$. $M_{31} = (23 \times 13) - (-6 \times -8) = 299 - 48 = 251$. The cofactor $C_{31} = (-1)^{3+1} \times M_{31} = (1) \times 251 = \textbf{251}$.

• For $a_{32} = 21$ (position (3,2)): The minor $M_{32}$ is the determinant of the submatrix formed by removing the third row and second column: $\left|\begin{array}{cc} 12 & -6 \\ 11 & 13 \end{array}\right|$. $M_{32} = (12 \times 13) - (-6 \times 11) = 156 - (-66) = 156 + 66 = 222$. The cofactor $C_{32} = (-1)^{3+2} \times M_{32} = (-1) \times 222 = \textbf{-222}$.

• For $a_{33} = 15$ (position (3,3)): The minor $M_{33}$ is the determinant of the submatrix formed by removing the third row and third column: $\left|\begin{array}{cc} 12 & 23 \\ 11 & -8 \end{array}\right|$. $M_{33} = (12 \times -8) - (23 \times 11) = -96 - 253 = -349$. The cofactor $C_{33} = (-1)^{3+3} \times M_{33} = (1) \times (-349) = \textbf{-349}$.

Therefore, the cofactors of the third-row entries are 251, -222, and -349.

Conclusion

We've successfully calculated all the cofactors for each row of matrix $A$! To summarize:

• The cofactors of the first-row entries are -393, -74, and 287.
• The cofactors of the second-row entries are -471, 222, and -91.
• The cofactors of the third-row entries are 251, -222, and -349.

Understanding how to compute cofactors is a fundamental skill in linear algebra. It not only helps in finding determinants and inverses but also lays the groundwork for more advanced concepts. Keep practicing these calculations, and you'll soon master them!

For further exploration and to deepen your understanding of matrix operations, you can visit Khan Academy's comprehensive section on linear algebra. They offer excellent resources and tutorials that can help you solidify your knowledge.