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Properties of Determinants

In this section we'll take a look at some properties of determinants that could make their computation simplified in some cases:

  1. The determinant of the identity matrix \(\det(I) = 1\).
  2. Swapping two rows/columns of a matrix \(A\) reverses the sign of the \(\det(A)\). More generally the sign of the determinant is given by \((-1)^{n} \times \det(A)\) where an odd number of swaps would result in a -ve sign and an even number in a +ve sign.

  3. Determinants behave like a linear function along a row or a column. This means that

    • Multiplying a row/column by a scalar \(\lambda\) would also scale the determinant by \(\lambda\). \[3 . \begin{vmatrix} 1 & 2 \\ 4 & 5 \end{vmatrix} = \begin{vmatrix} \mathbf{3} & \mathbf{6} \\ 4 & 5 \end{vmatrix} = \begin{vmatrix} 1 & \mathbf{6} \\ 4 & \mathbf{15} \end{vmatrix} = 3 \times -3 = -9 \] As a consequence of this for an \(n \times n\) matrix we also get that \(\det(\lambda A) = \lambda^{n} \det(A)\)
    • A determinant of the following form can be expressed as the sum of the determinants obtained by splitting the terms of a single row/column and retaining all the others as the same. \[\begin{vmatrix} a + a' & b + b' \\ c & d \\ \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} + \begin{vmatrix} a' & b' \\ c & d \\ \end{vmatrix} \]

    Note

    This however does not mean that \(\det(A + B) = \det(A) + \det(B)\). This would be a wrong interpretation!

  4. If two rows/columns of a matrix \(A\) are the same then \(det(A) = 0\)

  5. Determinants are invariant to elementary row or column operations. In other words adding or subtracting a scalar multiple of a row/column to/from another doesn't change the value of \(\det(A)\). \[\begin{vmatrix} 3 & 7 \\ -2 & 2 \\ \end{vmatrix} = \begin{vmatrix} 3 + 0.5(7) & 7 \\ -2 + 0.5(2) & 2 \\ \end{vmatrix} = \begin{vmatrix} 6.5 & 7 \\ -1 & 2 \\ \end{vmatrix} = 20 \]
  6. A matrix with one or more rows of zeros will have a determinant of 0.
  7. The determinant of an upper triangular matrix (also works for lower) ie., a matrix with all zeros below it's main diagonal is given by the product of it's diagonal elements, also called it's pivot elements. \[\begin{vmatrix} \rlap{\kern .24em 1}\raise .04em{\bigcirc} & 2 & 3 \\ 0 & \rlap{\kern .24em 5}\raise .04em{\bigcirc} & 6 \\ 0 & 0 & \rlap{\kern .24em 9}\raise .04em{\bigcirc} \end{vmatrix} = 1 \times 5 \times 9 = 45 \] As we'll see in later weeks, any (invertible) matrix can be reduced into it's corresponding upper triagular form using a method called Gauss elimination. Thus elimination is another way of finding determinants.
  8. A non-invertible matrix also known as a singular matrix has a zero determinant. Conversely a matrix \(A\) with \(\det(A) = 0\) is singular.
  9. If \(A\) is non-singular then \(\det(A^{-1}) = \frac{1}{\det(A)}\).
  10. The determinant of a matrix product is the product of the individual determinants, \(\det(AB) = \det(A)\det(B)\).
  11. Determinants are invariant to transpositions ie., \(\det(A^T) = \det(A)\).