Properties of the Determinant

• The determinant is often used to find if a square matrix is invertible. If the determinant of a square matrix  isequal to zero, the matrix is not invertible.

• If one row of an n x n triangular matrix is filled entirely with zeros,the determinant of that matrix is equal to zero.

• If two rows of a square matrix are equal or proportional to each otherthen the determinant of that matrix is equal to zero (see linear dependence).

• In a square matrix, the determinant of matrix is equal to the determinantof the transpose ofthe matrix or  is equalto .

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• The determinant of the product of two square matrices is equal to theproduct of the determinants of each matrix.

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• The determinant of the unit (identity) matrix is one.

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• The determinant changes sign when two rows are exchanged.

• Consider a matrix of order n x n and any scalar c, then the determinantof the scalar times the matrix is equal to the determinant of the matrixtimes the scalar raised to the n power (the order of the matrix).
  

Consider
c = 4

then
= 1(4) - 3(2)= -2

and
=  4(16)- 8(12) = -32

it can be found that
= 4²(-2) = 16(-2)= -32