Question

Exactly how many of the following properties are equivalent to the statement, For a given n>1, the n×n matrix A is invertible.

(i) det(A)≠0
(ii) The reduced matrix of A is the n×n identity

Answer 1

Both properties stated in the question, a nonzero determinant and reducibility to an identity matrix, are equivalent to an n×n matrix A being invertible for n>1, which means there are exactly two equivalent properties.

Step-by-step explanation:

The question asks for the properties that are equivalent to the statement: 'For a given n>1, the n×n matrix A is invertible'. The properties listed are:

1. det(A)≠0
2. The reduced matrix of A is the n×n identity matrix

Both of these properties are indeed equivalent to the matrix A being invertible. If a matrix has a nonzero determinant (Property i), it is a fundamental theorem of linear algebra that the matrix is invertible. Similarly, if a matrix can be reduced to the identity matrix through a series of elementary row operations (Property ii), this indicates that it has full rank and consequently an inverse exists. There are exactly two properties listed that are equivalent to the given statement about matrix A being invertible.