Question

If the statement is true, then answer True. If the statement is false, then give a counterexample to prove it is false.

Interchanging two rows of a given matrix changes the sign of its determinant.

Answer 1

The statement is true. Interchanging two rows of a given matrix results in a determinant of opposite sign, which is a fundamental property of determinants in linear algebra.

Step-by-step explanation:

The statement "Interchanging two rows of a given matrix changes the sign of its determinant" is true. This is a well-known property of determinants. The determinant of a matrix calculates the signed volume of the n-dimensional parallelepiped that the matrix's columns (or rows) span, and interchanging two rows (or two columns) of a matrix is like flipping the orientation of that volume, which changes the sign of the determinant.

To illustrate this with an example, consider a simple 2x2 matrix:

• Let matrix A be:
• | 1 2 |
• | 3 4 |
• Det(A) = 1*4 - 2*3 = 4 - 6 = -2

Now, if we interchange the two rows of A to get matrix B, we have:

• | 3 4 |
• | 1 2 |
• Det(B) = 3*2 - 4*1 = 6 - 4 = 2

Notice that Det(B) = -Det(A), demonstrating the change of sign when rows are interchanged.