Determinant of a Transformation Matrix (DP IB Applications & Interpretation (AI): HL): Revision Note

Determinant of a transformation matrix

What does the determinant of a transformation matrix represent?

• The absolute value of the determinant of a transformation matrix is the area scale factor of the transformation

  • Area scale factor = $\left|\det \mathit{T}\right|$

• The area of the image will be product of the area of the object and the absolute value of the determinant of the transformation matrix

  • $\mathrm{Area}\mathrm{of}\mathrm{image}=\left|\det \mathit{T}\right|\times \mathrm{Area}\mathrm{of}\mathrm{object}$

• The area is unchanged if $\left|\det \mathit{T}\right|=1$

  • The area increases if $\left|\det \mathit{T}\right|>1$

  • The area decreases if $\left|\det \mathit{T}\right|<1$

• If the determinant is negative then the orientation of the shape will be reversed

  • For example, if the shape has been reflected