Question

Is this true or false: two square matrices A and B can have the same determinant if and only if they are equal

Answer 1

Let’s use two 2×2 matrices, A and B, with elements a₁-a₄ and b₁-b₄ such that their determinants are a₁a₄-a₂a₃ and b₁b₄-b₂b₃. So when their determinants are equal a₁a₄-a₂a₃=b₁b₄-b₂b₃. Now, let a₁=1, a₂=2, a₃=3 and a₄=4 then |A|=-2.

Let b₁=7, b₂=5, b₃=13 and b₄=9, then |B|=-2. The determinants are equal but A≠B, and the statement is therefore false.