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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 the following.

\(2 \times 2\) 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 determinant of a \(2 \times 2\) matrix.

For \(A = \left[\begin{smallmatrix} 1 \amp 2 \\ -3 \amp 4 \end{smallmatrix}\right]\text{,}\) we have the following.

Another \(2 \times 2\) determinant calculation example

Subsection 8.4.2 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 8.4.3. Determinant of a \(3 \times 3\) matrix along a row.

Let's compute the determinant of the matrix from Discovery 8.7. 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 8.4.4. Determinant of a \(3 \times 3\) matrix along a column.
\(3 \times 3\) determinant calculation example revisited: choose a column

Again, 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 8.4.3 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 8.4.5. 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

However, the cofactor expansion along the chosen row will still involve two \(3\times 3\) minor determinant calculations.

\(4 \times 4\) determinant calculation example: expand

The minor determinants \(M_{31}\) and \(M_{33}\) will not be needed, since their corresponding entries are \(0\text{.}\) So we now have to chose a row or column in each of the remaining \(3 \times 3\) 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 \(2 \times 2\) crisscross pattern.

\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*}