Determinants (2×2 and 3×3)

The determinant is a scalar value associated with a square matrix. It encodes important geometric and algebraic information: whether a matrix is invertible, the area/volume scaling factor of a transformation, and more.

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The 2×2 Determinant

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$A=(ac​bd​):

$\det(A) = |A| = ad - bc$det(A)=∣A∣=ad−bc

Worked Example 1: $\det\begin{pmatrix} 3 & 2 \\ 1 & 5 \end{pmatrix} = 3(5) - 2(1) = 15 - 2 = 13$det(31​25​)=3(5)−2(1)=15−2=13

Worked Example 2: $\det\begin{pmatrix} 4 & 6 \\ 2 & 3 \end{pmatrix} = 4(3) - 6(2) = 12 - 12 = 0$det(42​63​)=4(3)−6(2)=12−12=0

When the determinant is zero, the matrix is called singular and has no inverse.

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The 3×3 Determinant

For $A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & k \end{pmatrix}$A=​adg​beh​cfk​​:

$\det(A) = a(ek - fh) - b(dk - fg) + c(dh - eg)$det(A)=a(ek−fh)−b(dk−fg)+c(dh−eg)

This is expansion along the first row using cofactors.

Worked Example 3: Find $\det\begin{pmatrix} 2 & 1 & 3 \\ 0 & -1 & 2 \\ 1 & 4 & -1 \end{pmatrix}$det​201​1−14​32−1​​.