If the matrix A is such that A = [2; 1; 7] * [1 9 5] (i.e., A is the outer product of the column vector [2, 1, 7]^T and the row vector [1, 9, 5]), then the determinant of A is equal to ______.

GATE 2014 · Engineering Mathematics · Determinant · easy

Answer: det(A) = 0

1. Compute the explicit matrix A: A = [[2*1, 2*9, 2*5],[1*1, 1*9, 1*5],[7*1, 7*9, 7*5]] = [[2,18,10],[1,9,5],[7,63,35]]
2. Identify linear dependence among rows: Row 1 = 2 * Row 2 and Row 3 = 7 * Row 2. Hence the three rows are linearly dependent.
3. Conclude determinant: rank(A) = 1 < 3 = n, so det(A) = 0.