Question

If \(A\) is an \(n \times n\) matrix, show that \(\operatorname{det} A=\) 0 if and only if some column of \(A\) is a linear combination of the other columns.

Short answer

\( \operatorname{det} A = 0 \) if and only if a column is a linear combination of others.

Step-by-step solution

Understand the determinant property

The determinant of a matrix, denoted as \( \operatorname{det} A \), provides information about the linear independence of the matrix's columns. Specifically, if \( \operatorname{det} A = 0 \), it indicates that the columns are linearly dependent; that is, one or more columns can be expressed as a linear combination of the others.

Establish the forward implication

We need to prove that if \( \operatorname{det} A = 0 \), then one of the columns of \( A \) is a linear combination of the other columns. If \( \operatorname{det} A = 0 \), it implies that the matrix \( A \) does not have full rank, meaning the rank is less than \( n \). In such a case, the columns are linearly dependent. Linear dependence directly results in one column being a linear combination of the others.

Establish the backward implication

Now, prove that if some column is a linear combination of the other columns, \( \operatorname{det} A = 0 \). Let’s assume that the last column of matrix \( A \), say \( \mathbf{v}_n \), can be expressed as a linear combination of the previous columns \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_{n-1} \) as \( \mathbf{v}_n = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_{n-1}\mathbf{v}_{n-1} \). When substituting \( \mathbf{v}_n \) in terms of the other columns, the rows of the matrix become linearly dependent, resulting in \( \operatorname{det} A = 0 \).

Conclusion based on both implications

By establishing both the forward and backward implications, we have shown that \( \operatorname{det} A = 0 \) if and only if one column of \( A \) is a linear combination of the other columns. This completes the proof.

Key concepts

Understanding Linear Dependence

Linear dependence is a vital concept in linear algebra that involves vectors. Vectors are said to be linearly dependent if one vector can be expressed as a combination of others with non-zero coefficients. This means that at least one vector in the set does not add anything new in terms of direction or dimension. It can be represented as a combination of other vectors.
For example, consider three vectors: \(\mathbf{v}_1, \mathbf{v}_2, \text{and } \mathbf{v}_3\). If \(\mathbf{v}_3\) can be expressed as \(a\mathbf{v}_1 + b\mathbf{v}_2\), where both \(a\) and \(b\) are not zero, then these vectors are linearly dependent.

• A key insight here is the connection between linear dependence and matrix determinants.
• If a matrix's columns are linearly dependent, then its determinant is zero.

Exploring Matrix Rank

The rank of a matrix is a measure of the dimension of the vector space spanned by its rows or columns. It tells us how many independent vectors are in the matrix, effectively representing the maximal number of linearly independent column or row vectors in the matrix.
If a matrix is \(n \times n\) and its rank is less than \(n\), it implies that not all columns are independent. This is where the determinant comes into play: a determinant of zero means that the matrix does not have full rank.

• A rank of less than \(n\) shows linear dependence among the columns.
• This lack of full rank results in the determinant being zero, revealing a key property of the matrix.

Decoding Linear Combinations

A linear combination involves taking a set of vectors and combining them using addition and scalar multiplication. For example, you might wonder, how can I express a vector \( \mathbf{v}\) using other vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots \mathbf{v}_n\)? The formula is \( \mathbf{v} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \ldots + c_n \mathbf{v}_n \), where \(c_1, c_2, \ldots, c_n\) are scalars.
This is fundamental to the idea of spanning a space. If you can write one vector as a linear combination of others, the vector doesn't add any 'new' direction to the span of the set.

• When dealing with matrices, if a column is a linear combination of other columns, it confirms the determinant is zero.
• This is because at least one column doesn't provide unique information, ensuring linear dependence.