Evaluating n x n Determinants with Minor and Cofactor Entries

Be sure to review what a Minor and Cofactor entry is, as this section will rely heavily on understanding these concepts.

Evaluating n x n Determinants Using Cofactors/Minors

Finding the determinant of a $2 \times 2$ matrix is relatively easy, however finding determinants for larger matrices eventually becomes tricker. We will look at two methods using cofactors to evaluate these determinants.

Cofactor Expansion Along a Row

We can evaluate the determinant of a larger matrix by selecting any row, multiplying each element in that row by their corresponding cofactor, and summing the results, that is for any row $k$:

(1)
\begin{align} \: \det(A) = \sum_{j=1}^n a_{kj}C_{kj} = a_{k1}C_{k1} + a_{k2}C_{k2} + ... + a_{kn}C_{kn} \end{align}

For example, suppose we have the following matrix $A = \begin{bmatrix}3 & 1 & 2\\ -1 & 2 & 0\\ 1 & 0 & 2 \end{bmatrix}$, and let's evaluating the determinant of $A$ by cofactor expansion along row 1. We thus get the formula $\det (A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$.

(2)
\begin{align} \det (A) = 3\begin{vmatrix} 2 & 0\\ 0 & 2\end{vmatrix} + 1\begin{vmatrix} -1 & 0\\ 1 & 2\end{vmatrix} + 2\begin{vmatrix} -1 & 2\\ 1 & 0\end{vmatrix} \\ \: \det (A) = 3[(2)(2) - (0)(0)] - 1[(-1)(2) - 0(1)] + 2[(-1)(0) - (2)(1)] \\ \det(A) = 3(4) - 1(-2) + 2(-2) \\ \det(A) = 12 +2 - 4 \\ \det(A) = 10 \end{align}
 Note: If we were select to do cofactor expansion along the second or third row, we would obtain the same answer for the determinant of $A$.

We were thus able to compute a $3 \times 3$ determinant by computing multiple $2 \times 2$ determinants. Note that if we wanted to compute a $4 \times 4$ determinant, we would have to break it up into a a bunch of $3 \times 3$ determinants, and work down to $2 \times 2$ again. As you can probably imagine, finding the determinants of extremely large matrices may be more difficult though possible now.

Cofactor Expansion Along a Column

If we choose, we can also find the determinant of a square matrix by doing the exact same method for cofactor expansion along a row but instead use columns. If done correctly, we will still obtain the same determinant with the following similar formula for cofactor expansion along a column for some column $k$:

(3)
\begin{align} \: \det (A) = \sum_{i=1}^m a_{1k}C_{ik} = a_{1k}a_{1k} + a_{2k}C_{2k} + ... + a_{mk}C_{mk} \end{align}

For example, let's consider the matrix from the earlier example and expand along the second column, that is:

(4)
\begin{align} \det (A) = a_{12}C_{12} + a_{22}C_{22} + a_{32}C_{32} \\ \det (A) = 1\begin{vmatrix} -1 & 0\\ 1 & 2\end{vmatrix} - 2\begin{vmatrix} 3 & 2\\ 1 & 2\end{vmatrix} + 0\begin{vmatrix} 3 & 2\\ -1 & 0\end{vmatrix} \\ \det (A) = -1[(-1)(2) - (0)(1)] + 2[(3)(2) - (2)(1)] + 0 \\ \det (A) = 2 +8 + 0 \\ \det (A) = 10 \end{align}

We note that once again we obtain the same determinant for $A$. Also note that we did not have to compute one of the cofactors, specifically $C_{32}$ since $a_{32} = 0$ and multiplication by $0$ results in $0$. Nothing these sort of elements is important in reducing work from calculating larger determinants.

Strategy for Cofactor Expansion

Considering the following matrix:

(5)
\begin{align} A = \begin{bmatrix} 2 & 0 & 1 & 4\\ 3 & -2 & 1 & 0 \\ 2 & 3& 1& 0\\ 1& 0 & 3 & 0 \end{bmatrix} \end{align}

We notice right away that column $4$ contains almost all zeroes. If we were to use cofactor expansion along the fourth column, we would greatly reduce the amount of work in terms of calculations as opposed to using cofactor expansion along let's say row $2$.

Always check for rows/columns with a large amount of zeroes as cofactor expansion along them is often easier. That said, you are not always guaranteed to see a row/column where cofactor expansion works best per say.