Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. Multiplication of an invertible matrix and its inverse is commutative.

Step-by-Step Solution

Answer

True

Step 1: Understanding the terms

Before solving this exercise, you need to be familiar with the key terms. An invertible matrix, also known as a nonsingular matrix or a nondegenerate matrix, is a square matrix that has an inverse. In other words, if \(A\) is an invertible matrix, there exists a matrix \(A^{-1}\) such that \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix. Now, the term 'commutative' refers to the property of certain operations that the result remains the same if the order of operands is reversed.

Step 2: Apply the definition

Let \(A\) be an invertible matrix and \(A^{-1}\) be its inverse, then according to the definition of the inverse of a matrix, \(AA^{-1} = A^{-1}A = I\). This tells that the order of multiplication doesn't affect the result, implying that multiplication of an invertible matrix and its inverse is commutative.

Step 3: Conclusion

So based on the definitions and working out the operation, we can conclude that the statement 'Multiplication of an invertible matrix and its inverse is commutative' is TRUE.