If the determinant of the matrix is −P, then what is the value of the…

For an n×n matrix, if every entry of one matrix is a constant multiple k of the corresponding entry in another matrix (that is, A = kB), the determinants are related by det(A) = kn · det(B) — scaling every entry by k scales each of the n rows by k, so the determinant scales by k raised to the number of rows.

1. Matrix B (the matrix whose determinant is given as −P) has entries [1, 3, 0; 2, 6, 4; −1, 0, 2], and matrix A (the matrix whose determinant is required) has entries [2, 6, 0; 4, 12, 8; −2, 0, 4].

2. Every entry of A is exactly twice the corresponding entry of B (2 = 2×1, 6 = 2×3, 4 = 2×2, 12 = 2×6, 8 = 2×4, −2 = 2×(−1), 4 = 2×2), so A = 2B with k = 2 and n = 3 (a 3×3 matrix).

3. Apply the scaling relationship: det(A) = 23 · det(B) = 8 · det(B).

4. Substitute det(B) = −P: det(A) = 8 × (−P) = −8P.

Cross-check by direct expansion: det(B) = 1(6·2 − 4·0) − 3(2·2 − 4·(−1)) + 0(2·0 − 6·(−1)) = 12 − 24 = −12, so P = 12 here. Expanding A directly gives det(A) = 2(12·4 − 8·0) − 6(4·4 − 8·(−2)) + 0(4·0 − 12·(−2)) = 96 − 192 = −96, which equals 8 × (−12) = −8P — confirming the scaling relationship.