Question

Use the theorem given below to show that if two rows of a square matrix are equal, then det A=0. The same is true for two columns. Why?

Let A be a square matrix.
a. If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A.
b. If two rows of A are interchanged to produce B, then det B = - det A.
c. If one row of A is multiplied by k to produce B, then det B = k det A.

Answer 1

Suppose that two rows of a square matrix A are equal. Without loss of generality, let the first two rows of A be equal. Then, we can subtract the first row from the second row to obtain a new matrix B, where the first two rows of B are zero and the remaining rows are the same as in A. Using the formula det B = det A from part a of the theorem, we get det B = det A, since subtracting one row from another does not change the determinant. However, we can also expand the determinant of B along the first row to get det B = 0, since the first two rows of B are zero. Therefore, we have det A = det B = 0, which shows that if two rows of a square matrix are equal, then the determinant is zero. The same argument can be applied to show that if two columns of a square matrix are equal, then the determinant is also zero.