Section I.3.1

Determinants: construction and basic properties

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Problem I.3.1.1

Write the skew-symmetric function Δ:R3→R\Delta : \mathbb {R}^{3} \rightarrow \mathbb {R} of three variables

Δ(x,y,z)=(y−x)(z−x)(z−y) \Delta (x, y, z)=(y-x)(z-x)(z-y)

as a third-order determinant.

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Problem I.3.1.2

Let A=(aij),A′=(aij′)A=\left(a_{i j}\right), A^{\prime }=\left(a_{i j}^{\prime }\right) be two n×nn \times n matrices, Δ,Δ′\Delta , \Delta^{\prime } their determinants. Compare Δ\Delta and Δ′\Delta^{\prime } in the cases:

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(a)

aij′=2i−jaija_{i j}^{\prime }=2^{i-j} a_{i j}

(b)

aij′=an+1−i,ja_{i j}^{\prime }=a_{n+1-i, j}

(c)

aij′=an+1−i,n+1−ja_{i j}^{\prime }=a_{n+1-i, n+1-j}.

Problem I.3.1.3

Show that

∣111…11121…11113…11111…n1111…1n+1∣=n!. \left|\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 & 1 \\ 1 & 2 & 1 & \ldots & 1 & 1 \\ 1 & 1 & 3 & \ldots & 1 & 1 \\ 1 & 1 & 1 & \ldots & n & 1 \\ 1 & 1 & 1 & \ldots & 1 & n+1 \end{smallmatrix}\right|=n ! .
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