When the determinant of an $$ n imes n $$ matrix is zero, we know At least one row is a linear combination of the others. Any linear combination of the $$ n $$ rows sums to zero. Exactly one row is a linear combination of the others. Every row is a linear combination of the others. Each row after the first one is a linear combination of the rows above it.

Question

When the determinant of an $$ n 	imes n $$ matrix is zero, we know At least one row is a linear combination of the others. Any linear combination of the $$ n $$ rows sums to zero. Exactly one row is a linear combination of the others. Every row is a linear combination of the others. Each row after the first one is a linear combination of the rows above it.

UpStudy AI · Accepted Answer

We are given an $$ n 	imes n $$ matrix with zero determinant. The key property of such a matrix is that its rows (or columns) are linearly dependent. This means there exists a set of scalars (not all zero) such that a linear combination of the rows is zero.

Let's examine each statement:

1. **At least one row is a linear combination of the others.**  
   Since the rows are linearly dependent, there exists at least one row that can be expressed as a linear combination of the other rows. This statement is **true**.

2. **Any linear combination of the $$ n $$ rows sums to zero.**  
   In general, any linear combination of the rows does not necessarily equal the zero row unless the coefficients are chosen in a specific way to yield the dependency relation. This statement is **false**.

3. **Exactly one row is a linear combination of the others.**  
   There is no restriction that only one row must be a combination of the others; in some cases, more than one row could be redundant. This statement is **false**.

4. **Every row is a linear combination of the others.**  
   It is not required that every row be expressible as a linear combination of the others; linear dependence only guarantees the existence of at least one such row. This statement is **false**.

5. **Each row after the first one is a linear combination of the rows above it.**  
   This is a specific structure (like what happens in row echelon form when there is a complete dependency) and is not guaranteed by the condition of having a zero determinant in general. This statement is **false**.

Thus, the only correct statement is the first one:

$$ \boxed{	ext{At least one row is a linear combination of the others.}} $$
At least one row is a linear combination of the others.

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