Minor and Cofactor Entries

# Minor Entries of a Matrix

We have already looking at a method for evaluating the determinant of a $2 \times 2$ square matrix $A = \begin{bmatrix} a & b\\ c & d \end{bmatrix}$, that is $\det(A) = ad - bc$. We will soon look at a method for evaluating the determinants of larger square matrices with what are known as minor entries and cofactors. For the time being, we will need to introduce what minor and cofactor entries are. First we will introduce a new notation for determinants:

(1)
\begin{align} \det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \end{align}

Of course this notation is not limited to $2 \times 2$ matrices as we will see.

 Definition: If $A$ is a square matrix of size $n \times n$, the Minor Entry $M_{ij}$ is the determinant of the $(n - 1) \times (n - 1)$ submatrix after deleting the $i^{th}$ row and $j^{th}$ column from $A$.

For example, consider the following $3 \times 3$ matrix:

(2)
\begin{align} A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} \end{align}

The minor entry $M_{11}$ is the determinant of the $2 \times 2$ matrix after deleting the 1st row and the 1st column. Therefore $M_{11} = \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix}$. We can calculate this determinant by the rule we used earlier, so $M_{11} = (5)(9) - (8)(6) = -3$. Therefore the minor entry $M_{11} = -3$.

If we wanted, we could calculate other minor entries. For example, let's calculate $M_{23}$, that is the determinant of the submatrix after deleting the second row and third column of $A$, $M_{23} = \begin{vmatrix} 1 & 2 \\ 7 & 8 \end{vmatrix} = 1(8) - 7(2) = -6$.

# Cofactors Entries of a Matrix

 Definition: Given that $M_{ij}$ is the minor entry for row $i$ and column $j$ in a square matrix $A$, the Cofactor of row $i$ and column $j$ is denoted as $C_{ij}$ and is calculated by the formula $C_{ij} = (-1)^{i+j} M_{ij}$.

We note that if the sum $i + j$ is even, then $C_{ij} = M_{ij}$, and that if the sum is odd, then $C_{ij} = -M_{ij}$. Hence, the only difference between the related minor entries and cofactors may be a sign change or nothing at all. Whether $C_{ij} = M_{ij}$ or $C_{ij} = -M_{ij}$ has a pattern for square matrices as illustrated: For example $C_{12} = -M_{12}$. Of course if you forget, you can always use the formula $C_{ij} = (-1)^{i+j} M_{ij}$, for example $C_{12} = (-1)^{1+2} M_{ij} = (-1)^3 M_{ij} = -M_{ij}$.

Now let's try an actual example by finding $C_{31}$ given the following matrix:

(3)
\begin{align} A = \begin{bmatrix} 3 & 3 & 3\\ -5 & 6 & 5\\ 2 & 9 & 8 \end{bmatrix} \end{align}

After deleting the 3rd row and the 1st column, we find $M_{31}$ by taking the determinant of this submatrix, that is $M_{31} = \begin{vmatrix}3 & 3\\ 6 & 5\end{vmatrix} = (3)(5) - (3)(6) =-3$ We then note that $i + j = 3 + 1 = 4$ which is even, so $M_{ij} = C_{ij} = -3$, so the cofactor entry of row $3$ column $1$ is $-3$.