Problem 60 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I use matrices to solve linear systems, the only arithmetic involves multiplication or a combination of multiplication and addition.

Step-by-Step Solution

Answer

The statement partially makes sense because while multiplication and a combination of multiplication and addition are involved in solving linear systems using matrices, subtraction and indirect division (multiplication by the reciprocal) are also necessary.

Step 1: Understand the statement

The statement is suggesting that the process of solving linear systems using matrices only involves two types of arithmetic: multiplication, or a combination of multiplication and addition.

Step 2: Reflect on the process of solving linear systems using matrices

When we solve linear systems using matrices, we use row operations. Those operations are: (1) Swapping two rows (2) Multiplying a row by a non-zero constant (3) Adding a multiple of one row to another row. Clearly, alongside multiplication and addition, subtraction (when we subtract one row from another) is also a common operation. Furthermore, division is, indirectly, involved when we multiply a row by a non-zero constant (which is equivalent to multiplying by the reciprocal aka division).

Step 3: Evaluate the statement

With the process of solving linear systems using matrices in mind, it is evident that the statement does not make complete sense. Yes, multiplication and addition are indeed involved, but solving matrices would be incomplete without subtraction and division.