A=diag(3,1). Which eigenvalue corresponds to eigenvector e1=(1,0)?
Matrix as a transformationA = [3 0; 0 1], det A = 3
original gridimage grid under Aimage of the unit square, area 3î ↦ (3, 0)ĵ ↦ (0, 1)
A sends î to (3, 0) and ĵ to (0, 1); the faint original grid becomes the gold image grid. The unit square (area 1) becomes the parallelogram of area |det A| = 3 — the size of the determinant is exactly that area factor.