Let a be a third-order matrix, A1, A2, A3 are all three-dimensional vectors, satisfying that | A1, A2, A3 | is not equal to 0, Ask for a

Let a be a third-order matrix, A1, A2, A3 are all three-dimensional vectors, satisfying that | A1, A2, A3 | is not equal to 0, Ask for a

A(a1,a2,a3)=(a1+a2,-a1+2a2-a3,a2-3a3)=(a1,a2,a3)K
K=
1 -1 0
1 2 1
0 -1 -3
Take determinants on both sides of the equation, because | A1, A2, A3 ≠ 0, so
|A| = |K| = -8.