Is an invertible matrix always lin. independent?

In summary, an invertible nxn matrix guarantees that the equation Ax=b is consistent for each b and also implies that the matrix's determinant is non-zero. This is because if a matrix is invertible, its rows or columns, thought of as vectors, are independent.
  • #1
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Because according to my book,

If A's an invertible nxn matrix, then the eq. Ax=b is consistent for EACH b...

this consistency would imply lin. independency?
 
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  • #2
Yes. Also, the determinant of an invertible matrix is non-zero. The determinant of a matrix of linearly dependent vectors is 0.
 
  • #3
Strictly speaking, the term "independent" applies to sets of vectors, not matrices so it does not make sense to ask if a matrix is independent. What you are really asking is whether the rows or columns of the matrix, thought of as vectors are independent. The answer to that is yes. If a matrix is invertible, then its rows (or columns), thought of as a set of vectors, is indepependent.
 

1. What is an invertible matrix?

An invertible matrix, also known as a nonsingular matrix, is a square matrix whose determinant is not equal to zero. This means that the matrix has an inverse, which is another matrix that, when multiplied by the original matrix, results in the identity matrix.

2. What does it mean for a matrix to be linearly independent?

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the other vectors. In the context of matrices, this means that none of the columns can be expressed as a linear combination of the other columns.

3. Is an invertible matrix always linearly independent?

No, an invertible matrix is not always linearly independent. A matrix can be invertible but still have linearly dependent columns, meaning that one or more columns can be expressed as a linear combination of the other columns.

4. What is the relationship between invertibility and linear independence?

An invertible matrix must have linearly independent columns, but the reverse is not necessarily true. This means that if a matrix is invertible, it must also be linearly independent, but a matrix can be linearly independent without being invertible.

5. Are there any exceptions to the rule that an invertible matrix must be linearly independent?

Yes, there are exceptions. For example, a square matrix with only one non-zero column will be linearly independent but not invertible, as its determinant will be equal to zero. Similarly, a square matrix with only one non-zero row will be linearly independent but not invertible.

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