Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. When the product of two square matrices is the identity matrix, the matrices are inverses of one another.

Step-by-Step Solution

Answer

False, the statement is incomplete. It's necessary, but not sufficient, for the product of two matrices to yield the identity matrix. A necessary and sufficient condition is that the product yields the identity matrix, regardless of the order of multiplication.

Step 1: Understand the definitions

Two primary definitions need to be understood in this problem, the 'Identity Matrix' and 'Matrix Inverse'. An identity matrix is a square matrix with 1s on the diagonal and 0s everywhere else. A matrix \(A\) is considered the inverse of a matrix \(B\) if their product, in any order, results in the identity matrix.

Step 2: Analyze the statement

Now let's apply these definitions to our problem. If the product of two square matrices \(A\) and \(B\) is an identity matrix, it means \(AB = I \). According to the definition of inverses, \(A\) and \(B\) are inverses only if \(AB = BA =I \). Based on this, the statement in the problem is incomplete.

Step 3: Deduce if the statement is true or false

From the above analysis, we know that the necessary condition for \(A\) and \(B\) to be inverses of each other is not only that their product equals the identity matrix, but also that the product remains the identity regardless of the order of multiplication. Because the statement doesn't specify this additional condition about commutativity, the statement is false.