Section 8.4 Examples
Subsection 8.4.1 Determinants of \(2 \times 2\) matrices
An easy way to remember the \(2 \times 2\) determinant formula is with a crisscross pattern, as illustrated below for general \(2 \times 2\) matrix \(A = \left[\begin{smallmatrix} a \amp b \\ c \amp d \end{smallmatrix}\right]\text{.}\)
Determinant calculation pattern for \(2 \times 2\) matrices
Example 8.4.1. Determinant of a \(2 \times 2\) matrix.
For \(A = \left[\begin{smallmatrix} 1 \amp 2 \\ 3 \amp 4 \end{smallmatrix}\right]\text{,}\) we have
\(2 \times 2\) determinant calculation example
Watch out for double negatives! The next example illustrates the occurrence of a double negative in a determinant calculation.
Example 8.4.2. Another \(2 \times 2\) determinant.
For \(A = \left[\begin{smallmatrix} 1 \amp 2 \\ -3 \amp 4 \end{smallmatrix}\right]\text{,}\) we have
Another \(2 \times 2\) determinant calculation example
Subsection 8.4.2 Minors and cofactors of \(3 \times 3\) matrices
Subsubsection 8.4.2.1 Minors
A minor determinant is just a one-size-smaller determinant. To obtain that smaller matrix, we remove one row and one column. Usually we specify which to remove by focusing on a single entry and removing the row and column that contain the entry.
Example 8.4.3. Minor determinants in a \(3 \times 3\) matrix.
Let’s compute a couple of minor determinants in the matrix from Discovery 8.7:
\begin{equation*}
\left[\begin{array}{rrr}
3 \amp 1 \amp 0 \\
-2 \amp -2 \amp 1 \\
0 \amp 1 \amp -1
\end{array}\right]\text{.}
\end{equation*}
The notation \(M_{11}\) means the minor associated to the \((1,1)\) entry, so we should remove both the first row and the first column, leaving behind a \(2 \times 2\) matrix.
\(3 \times 3\) minor determinant calculation example: remove associated row and column
We can now compute this minor determinant using the \(a d - b c\) pattern for \(2 \times 2\) determinants.
\(3 \times 3\) minor determinant calculation example: calculate associated \(2 \times 2\) determinant
Now let’s try the \(M_{23}\) minor determinant. This time we should remove the second row and the third column.
\(3 \times 3\) minor determinant calculation example: remove associated row and column
Again, from here we compute this minor determinant using the \(a d - b c\) pattern.
\(3 \times 3\) minor determinant calculation example: calculate associated \(2 \times 2\) determinant
Subsubsection 8.4.2.2 Cofactors
A cofactor just takes a minor determinant and (sometimes) flips its sign: when the corresponding entry is at an “even” position then the cofactor is equal to the minor determinant value, and when the corresponding entry is at an “odd” position then the sign is flipped.
Example 8.4.4. Cofactors in a \(3 \times 3\) matrix.
Let’s continue Example 8.4.3 above. The minor determinant \(M_{11}\) corresponds to the \((1,1)\) entry in the matrix, which is at an “even” position since \(1 + 1 = 2\) is even. So the corresponding cofactor value is equal to the minor determinant value:
\begin{equation*}
C_{11} = M_{11} = 1 \text{.}
\end{equation*}
But the minor determinant \(M_{23}\) corresponds to the \((2,3)\) entry in the matrix, which is at an “odd” position since \(2 + 3 = 5\) is odd. So the corresponding cofactor value is equal to the negative of the minor determinant value:
\begin{equation*}
C_{23} = - M_{23} = -3 \text{.}
\end{equation*}
Subsection 8.4.3 Determinants of \(3 \times 3\) matrices
For a \(3 \times 3\) matrix, we choose a single row or column and perform a cofactor expansion. It’s usually best to choose the row or column with the most zeros, since for a zero entry the “entry times cofactor” part of the expansion for that entry will be zero no matter the value of the cofactor, and we don’t actually have to calculate that cofactor. Also, we will use our cofactor sign patterns from Subsection 8.3.4 (see Pattern (8.3.1)), instead of calculating \((-1)^{i+j}\) explicitly.
Example 8.4.5. Determinant of a \(3 \times 3\) matrix along a row.
Let’s compute the determinant of the matrix from Discovery 8.7:
\begin{equation*}
\left[\begin{array}{rrr}
3 \amp 1 \amp 0 \\
-2 \amp -2 \amp 1 \\
0 \amp 1 \amp -1
\end{array}\right]\text{.}
\end{equation*}
Any of the first row or column or the third row or column would be good choices as they all contain a zero entry. Let’s choose the third row, since it also contains some \(1\)s, which will simplify things a bit. Notice how we have also annotated that row with the cofactor sign pattern.
\(3 \times 3\) determinant calculation example: choose a row
Now expand along that third row.
\(3 \times 3\) determinant calculation example: expand
The minus sign between the first two terms in the expansion is the proper cofactor sign for the middle entry of the third row. Also, recall that a cofactor for an entry involves the minor for that entry — the determinant of the smaller matrix obtained by removing the row and column in which that entry sits. We have indicated each removal of a row or column by a strike-through. Since \(A\) is \(3 \times 3\text{,}\) all of its minors are \(2 \times 2\) determinants that we can compute with our crisscross pattern. However, since the \((3,1)\) entry is \(0\text{,}\) there is no need to compute the \((3,1)\) minor.
\(3 \times 3\) determinant calculation example: reduce to \(2 \times 2\) minors
Using our crisscross pattern for \(2 \times 2\) determinants, we can now compute
\begin{align*}
\det A
\amp = 0 - 1 \cdot \bigl[3\cdot 1 - 0 \cdot (-2)\bigr] + (-1) \cdot \bigl[3\cdot(-2) - 1\cdot(-2)\bigr]\\
\amp = -3 + (-1)(-4)\\
\amp = 1.
\end{align*}
Just to check, let’s compute the determinant in the above example again using a cofactor expansion along the second column.
Example 8.4.6. Determinant of a \(3 \times 3\) matrix along a column.
Let’s again compute the determinant of the matrix from Discovery 8.7, but this time along the middle column.
\(3 \times 3\) determinant calculation example revisited: choose a column
Expand along the chosen column.
\(3 \times 3\) determinant calculation example revisited: expand
In the expansion, the negative sign in front of the first term and the minus sign between the second and third terms are from the cofactor sign pattern for the second column.
Now reduce to a combination of \(2 \times 2\) determinants.
\(3 \times 3\) determinant calculation example revisited: reduce to \(2 \times 2\) minors
Apply the \(2 \times 2\) crisscross pattern.
\begin{align*}
\det A
\amp = (-1)(2-0) + (-2)(-3-0) - 1 \cdot (3-0)\\
\amp = -2 + 6 - 3\\
\amp = 1.
\end{align*}
In the end, we got the same result as our first calculation, which is not a coincidence — see Theorem 8.5.1.
Subsection 8.4.4 Minors and cofactors of \(4 \times 4\) matrices
Applying the one-size-smaller pattern again, a minor determinant in a \(4 \times 4\) matrix is the determinant of a \(3 \times 3\) matrix obtained by removing one row and one column. And again cofactor values are equal to minor determinant values, except that we flip the signs for values associated to “odd” positions with the \(4 \times 4\) matrix.
Example 8.4.7.
Consider the matrix
\begin{equation*}
\left[\begin{array}{rrrr}
-1 \amp 3 \amp 1 \amp 0 \\
-5 \amp 6 \amp 7 \amp 8 \\
2 \amp -2 \amp -2 \amp 1 \\
2 \amp 0 \amp 1 \amp -1
\end{array}\right]\text{.}
\end{equation*}
To compute the \(M_{21}\) minor determinant, we remove the second row and the first column.
\(4 \times 4\) minor determinant calculation example: remove associated row and column
You might recognize this \(3 \times 3\) matrix as the same as the one from the examples in Subsection 8.4.3, so we already know its determinant. Also, the \((2,1)\) entry in the original \(4 \times 4\) matrix is in an “odd” position since \(2 + 1 = 3\) is odd, so must flip the sign to obtain the \(C_{21}\) cofactor value from the \(M_{21}\) minor determinant value:
\begin{align*}
M_{21} \amp = 1 \text{,} \amp C_{21} \amp = - M_{21} = -1 \text{.}
\end{align*}
Subsection 8.4.5 Determinants of \(4 \times 4\) matrices
Finally, here is a \(4 \times 4\) example. We’ll do one with a few zeros, so that it doesn’t get too out of hand.
Example 8.4.8. Determinant of a \(4 \times 4\) matrix.
Consider
\begin{equation*}
A = \left[\begin{array}{rrrr}
1 \amp -1 \amp 2 \amp 1 \\
2 \amp 0 \amp 1 \amp 1 \\
0 \amp 1 \amp 0 \amp -3 \\
1 \amp -2 \amp -1 \amp 0
\end{array}\right].
\end{equation*}
Let’s choose the third row, as that has two zero entries.
\(4 \times 4\) determinant calculation example: choose a row
The cofactor expansion along the chosen row will involve only two \(3 \times 3\) minor determinant calculations — minor determinants \(M_{31}\) and \(M_{33}\) will not be needed, since their corresponding entries are \(0\text{.}\)
\(4 \times 4\) determinant calculation example: expand
Next we choose a row or column in each of the remaining minor determinants.
\(4 \times 4\) determinant calculation example: reduce to \(3 \times 3\) minors
Notice how the cofactor signs in the chosen row/column follow the \(3 \times 3\) pattern, not the \(4 \times 4\) pattern from the original matrix.
Now expand each of these \(3 \times 3\) minor determinants.
\(4 \times 4\) determinant calculation example: expand \(3 \times 3\) minors
Now reduce to a combination of \(2 \times 2\) determinants.
\(4 \times 4\) determinant calculation example: reduce to \(2 \times 2\) minors
Finally, we can apply the \(2 \times 2\) criss-cross pattern as illustrated above.
\begin{align*}
\det A
\amp = -\bbrac{(-2-1)-(-1-2)} + 3\Bbrac{-2\bbrac{1-(-4)}-\bbrac{-2-(-1)}}\\
\amp = -(-3+3) + 3(-10 + 1)\\
\amp = -27.
\end{align*}
