Question

Without changing their meanings, convert each of the following sentences into a sentence having the form "If \(P\), then \(Q .\) " A matrix is invertible provided that its determinant is not zero.

Short answer

The sentence 'A matrix is invertible provided that its determinant is not zero' can be rewritten as 'If the determinant of a matrix is not zero, then the matrix is invertible'.

Step-by-step solution

- Identify the 'P' and 'Q'

In the provided sentence, 'its determinant is not zero' is the necessary condition providing the possibility for a matrix to be invertible. This is the 'If' part. So, 'P' is 'the determinant of a matrix is not zero'. Conversely, 'A matrix is invertible' is the condition that follows or results from the determinant not being zero. This is the 'then' part, so 'Q' is 'the matrix is invertible'.

- Form the 'If P, then Q' statement

Now that 'P' and 'Q' are determined, substitute these into the 'If P, then Q' structure. The sentence becomes: 'If the determinant of a matrix is not zero, then the matrix is invertible'.