If two rows/columns are identical then the determinant isQuestion 9Answera.1b.1/2c.-1d.Zero

Break Down the Problem

1. Understand what a determinant is in the context of matrices.
2. Identify the implications of having two identical rows or columns in a matrix.

Relevant Concepts

3. The determinant of a matrix is a scalar value that can be computed from its elements and encapsulates certain properties of the matrix.
4. A key property of determinants is that if a matrix has two identical rows or columns, then its determinant is zero.

Analysis and Detail

5. If we denote a matrix $A$ with rows $\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_n$, having $\mathbf{r}_i = \mathbf{r}_j$ for any $i \neq j$ means that the rows are not linearly independent. Thus, the volume interpretation of the determinant (as a measure of how much the transformation defined by the matrix scales n-dimensional space) leads us to a volume of zero, indicating that the rows do not span an n-dimensional space.

Verify and Summarize

6. Therefore, the determinant of any matrix that has two identical rows or columns will always be:

$\text{det}(A) = 0$

Final Answer

d. Zero