Determinant basics: the \(2\times 2\) formula and what it means

Key idea

For a \(2\times 2\) matrix \[ A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}, \] the determinant is \[ \det(A)=ad-bc. \] This number tells you whether the transformation is invertible (it is invertible iff \det(A)≠ 0), and in 2D it represents the signed area scaling factor. A negative determinant means orientation is reversed (a “flip”).

Worked example

Try it

Summary

• \(\det\!\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\).
• \(\det(A)=0\) means \(A\) is singular (not invertible).

\(3\times 3\) determinants: cofactor expansion and smart choices

Key idea

For a \(3\times 3\) matrix, a reliable method is cofactor (Laplace) expansion. The sign pattern is: \[ \begin{pmatrix} + & - & +\\ - & + & -\\ + & - & + \end{pmatrix}. \] Expanding along a row or column with zeros reduces the work a lot, because any term with a \(0\) entry disappears.

Worked example

Try it

Summary

• Use cofactor expansion, and choose a row/column with zeros when possible.
• Repeated or proportional rows/columns \(\Rightarrow \det(A)=0\).

Row operations and determinant properties

Key idea

These three row operations have predictable effects on determinants:

• Swap two rows: multiplies the determinant by \(-1\).
• Multiply a row by \(k\): multiplies the determinant by \(k\).
• Add \(k\) times one row to another: does not change the determinant.

This lets you use elimination to create zeros and then compute the determinant of a triangular matrix as a product of diagonal entries.

Worked example

Try it

Summary

• Track row swaps and row scalings; use row-addition operations freely (they do not change \(\det\)).
• Triangular matrix \(\Rightarrow\) determinant is the product of diagonal entries.

Diagonal, triangular, and permutation matrices

Key idea

• Diagonal matrix: \(\det(\mathrm{diag}(d_1,\dots,d_n))=d_1\cdots d_n\).
• Triangular matrix: for upper/lower triangular matrices, \(\det\) is the product of diagonal entries.
• Permutation matrix: \(\det(P)=+1\) for an even permutation and \(\det(P)=-1\) for an odd permutation.

Worked example

Try it

Summary

• Diagonal/triangular matrices: determinant is the product of the diagonal entries.
• Permutation matrices: determinant is \(\pm 1\) depending on parity (even/odd permutation).

Determinant algebra: \(\det(AB)\), \(\det(kA)\), and \(\det(A^{-1})\)

Key idea

• Product rule: \(\det(AB)=\det(A)\det(B)\).
• Scalar multiple: if \(A\) is \(n\times n\), then \(\det(kA)=k^n\det(A)\).
• Inverse: if \(A\) is invertible, then \(\det(A^{-1})=\dfrac{1}{\det(A)}\).
• Transpose: \(\det(A^T)=\det(A)\).

Worked example

Try it

Summary

• \(\det(AB)=\det(A)\det(B)\).
• \(\det(kA)=k^n\det(A)\) for \(n\times n\) matrices.

When is \(\det(A)=0\)? Fast checks for singular matrices

Key idea

• If two rows (or columns) are equal, then \(\det(A)=0\).
• If one row (or column) is a multiple of another, then \(\det(A)=0\).
• If a row or column is all zeros, then \(\det(A)=0\).
• \(\det(A)=0\) means the rows/columns are linearly dependent, so \(A\) is not invertible.

Worked example

Try it

Summary

• Dependent rows/columns \(\Rightarrow \det(A)=0\).
• Scaling each row by \(k\) scales the determinant by \(k\) each time.

Why determinants matter

Where determinants show up

• Invertibility: \(A\) is invertible iff \det(A)≠ 0.
• Area and volume: \(|\det(A)|\) is an area/volume scaling factor.
• Solving systems: determinants appear in Cramer’s rule and formulas involving inverses.
• Eigenvalues: the determinant equals the product of eigenvalues (counting multiplicity) for square matrices.

Worked example: area of a parallelogram

Try it

Final recap

• \(2\times 2\): \(\det\!\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\).
• \(3\times 3\): use cofactor expansion; pick rows/columns with zeros when possible.
• Row operations: swap \(\Rightarrow\) sign flip; scale a row \(\Rightarrow\) scale \(\det\); row-addition \(\Rightarrow\) no change.
• Triangular/diagonal: determinant is the product of diagonal entries.
• Rules: \(\det(AB)=\det(A)\det(B)\), \(\det(kA)=k^n\det(A)\), \det(A)≠ 0 \(\Leftrightarrow\) invertible.