Determinants

Principle

The determinant is a number assigned to a square matrix. Determinants are defined only for square matrices, and they help decide whether a square linear system has a unique solution.

For a square matrix \(A\), a nonzero determinant means \(A\mathbf{x}=\mathbf{b}\) has a unique solution for every compatible vector \(\mathbf{b}\).

Notation

• The determinant of \(A\) is written \(\det(A)\) or \(|A|\).
• A \(2\times2\) matrix \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) has determinant \(ad-bc\).
• The minor \(M_{ij}\) is the determinant of the matrix left after deleting row \(i\) and column \(j\).
• The cofactor signs follow the checkerboard pattern \(+,-,+,-,\ldots\).

Method

To compute a determinant by expanding along the first row:

1. Confirm that the matrix is square.
2. For each entry in the first row, delete its row and column to form its minor.
3. Apply alternating signs \(+,-,+,\ldots\) across the first row.
4. Multiply each first-row entry by its signed minor.
5. Add the signed terms.

Rules

• Determinants apply to square matrices only.
• For \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\), \(\det(A)=ad-bc\).
• For a \(3\times3\) matrix, first-row cofactor expansion is

\[ \det(A)=a_{11}M_{11}-a_{12}M_{12}+a_{13}M_{13}. \]

• If \(\det(A)\ne0\), then the square system \(A\mathbf{x}=\mathbf{b}\) has a unique solution.
• If \(\det(A)=0\), then the square system does not have a unique solution.

Examples

For a \(2\times2\) matrix,

\[ \det\begin{pmatrix}3&5\\2&7\end{pmatrix} =3\cdot7-5\cdot2=21-10=11. \]

Now compute the determinant of

\[ A=\begin{pmatrix} 2&1&3\\ 0&-1&4\\ 5&2&1 \end{pmatrix}. \]

Expand along the first row:

\[ \det(A)=2M_{11}-1M_{12}+3M_{13}. \]

The first minor is

\[ M_{11}=\det\begin{pmatrix}-1&4\\2&1\end{pmatrix} =(-1)(1)-4(2)=-1-8=-9. \]

The other two minors are

\[ M_{12}=\det\begin{pmatrix}0&4\\5&1\end{pmatrix}=0(1)-4(5)=-20, \]

and

\[ M_{13}=\det\begin{pmatrix}0&-1\\5&2\end{pmatrix}=0(2)-(-1)(5)=5. \]

Therefore

\[ \det(A)=2(-9)-1(-20)+3(5)=-18+20+15=17. \]

Because \(\det(A)\ne0\), the square system \(A\mathbf{x}=\mathbf{b}\) has a unique solution for every compatible \(\mathbf{b}\).

Checks

• Check that the matrix is square before trying to take its determinant.
• Check the \(2\times2\) formula order: \(ad-bc\), not \(ab-cd\).
• Check cofactor signs carefully: first row uses \(+,-,+\).
• Check that each minor deletes exactly one row and one column.
• Check that a nonzero determinant means a unique solution for a square system.