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Determinants and singular matrices

A zero determinant identifies a square matrix with no inverse.

For [[a,b],[c,d]], the determinant is ad − bc. In a two-dimensional transformation, its absolute value is the area scale factor. A negative determinant reverses orientation.

A singular matrix maps distinct inputs onto the same output and cannot be undone by an inverse. For a parameterised matrix, solve det(A) = 0 to find the exceptional values. Do not divide by its determinant before excluding those values.

Worked example

For A = [[k,2],[3,4]], find when A is singular.

  1. det(A) = 4k − 6.
  2. Set 4k − 6 = 0.

Answer: k = 3/2

Revise first: Matrix multiplication.

Practise determinants and singular matrices

Course mapping

These specification references show where the topic occurs. The questions cover only some parts of each topic.

Next practice: Inverse matrices.