Section I.1.4

Determinants of low orders

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Problem I.1.4.1

Formula (8) is easier to remember if one uses the following visual rule of signs for writing out the products occurring in the expansion of a third-order determinant:

Find an analogous rule for a fourth-order determinant.

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Problem I.1.4.2

Show that all six terms in the expansion of a third-order determinant cannot be simultaneously positive.

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Problem I.1.4.3

Verify that

∣a11a12a13a21a22a23a31a32a33∣=∣a11a21a31a12a22a32a13a23a33∣,∣0ab−a0c−b−c0∣=0 \left|\begin{smallmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{smallmatrix}\right| =\left|\begin{smallmatrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} \end{smallmatrix}\right|, \qquad \left|\begin{smallmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{smallmatrix}\right| = 0
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