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Linear algebra

Determinants

The determinant tells you whether a matrix is invertible. Learn both how to compute it and how its properties simplify the work.

Basic, Standard, Advanced: University Year 1 · 1st semester

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2×2 and 3×3 determinants (Sarrus' rule)

Formulas
∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc
∣a11a12a13a21a22a23a31a32a33∣=a11a22a33+a12a23a31+a13a21a32−a13a22a31−a11a23a32−a12a21a33\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}

For 3×33 \times 3, add the three down-right diagonal products and subtract the three down-left ones (Sarrus' rule). This does not work for 4×44 \times 4 or larger.

Cofactor expansion and row operations

Expanding along row ii: ∣A∣=∑j(−1)i+jaij∣Aij∣|A| = \sum_j (-1)^{i+j}a_{ij}|A_{ij}|, where AijA_{ij} deletes row ii and column jj. Expand along the row or column with the most zeros.

For 4×44 \times 4 and larger, use row operations to create zeros first, or reduce to a triangular matrix, whose determinant is the product of the diagonal entries.

Properties of determinants

Properties
  • Swapping two rows changes the sign.
  • Multiplying one row by kk multiplies the determinant by kk (so ∣kA∣=kn∣A∣|kA| = k^n|A| for n×nn \times n).
  • Adding a multiple of one row to another changes nothing.
  • ∣AB∣=∣A∣∣B∣|AB| = |A||B|, ∣tA∣=∣A∣|{}^tA| = |A| and ∣A−1∣=1∣A∣|A^{-1}| = \frac{1}{|A|}.

The same rules hold for columns. To factor a determinant with letters, use these rules to pull out a common factor.

Cramer's rule, areas and volumes

If ∣A∣≠0|A| \neq 0, the solution of Ax=bA\boldsymbol{x} = \boldsymbol{b} is xj=∣Aj∣∣A∣x_j = \frac{|A_j|}{|A|}, where AjA_j replaces column jj of AA by b\boldsymbol{b} (Cramer's rule).

The parallelogram spanned by two vectors has area ∣ad−bc∣|ad - bc|, and the parallelepiped spanned by three vectors has volume equal to the absolute value of their determinant.