Problem 84 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Two \(2 \times 2\) invertible matrices can have a matrix sum that is not invertible.

Step-by-Step Solution

Answer

The statement 'Two \(2 \times 2\) invertible matrices can have a matrix sum that is not invertible' is conditionally true. It is not true or false universally, as it depends on the specific matrices.

Step 1: Definition of Invertible Matrices

An invertible matrix (also known as a non-singular matrix or a non-degenerate matrix) is a square matrix that has an inverse. In other words, if \(A\) is a matrix, there exists a matrix \(A^{-1}\) such that \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix of the same order.

Step 2: Adding Invertible Matrices

Let's denote our two \(2 \times 2\) invertible matrices as \(A\) and \(B\). Their sum is represented as \(C=A+B\). The elements of \(C\) will be the sum of corresponding elements of \(A\) and \(B\).

Step 3: Checking Invertibility of the Sum

The statement given implies that the sum of two invertible matrices, \(C=A+B\), is not invertible. However, this is not necessarily true, because the sum of two invertible matrices can still be invertible, or it may not be invertible. It actually depends on the specific matrices \(A\) and \(B\). Therefore, the statement is not definitively true or false as it depends on the specific matrices. It's important to note, however, that generally speaking, the operations applied to matrices, such as addition in this case, do not necessarily preserve the property of invertibility.