Question

which of the following sets of data would you likely represent with a square matrix?
a shoe sales person wants to use an augmented matrix to represent the quantity of childrens shoes and adult shoes sold at his store for two days.
a racquetball club has a tournament where each member plays every other member and represents the scores in a matrix.
a student wants to write a coefficient matrix representing a linear system with two equations and three unknowns.
a teacher wants to write an augmented matrix representing a linear system with four equations and four unknowns.

Explanation:

Step1: Recall the definition of a square matrix

A square matrix has the same number of rows and columns.

Step2: Analyze each option

• For the shoe salesperson: There are 2 types of shoes (children's and adult's) and 2 days. The matrix would be \(2\times2\) (augmented matrix for 2 variables and 2 equations). But an augmented matrix for a system of equations (if we consider it as a system) would have an extra column. However, the key is the coefficient part. But let's check others.
• For the racquetball club: If there are \(n\) members, the matrix representing scores where each member plays every other member is a \(n\times n\) matrix. But it's not a matrix for a linear - system - related context in the strict sense of square matrices for linear algebra (where we relate to systems of equations).
• For the student with 2 equations and 3 unknowns: The coefficient matrix is \(2\times3\), which is not square.
• For the teacher with 4 equations and 4 unknowns: The coefficient matrix of the linear system is \(4\times4\) (a square matrix). And an augmented matrix for this system is \((4\times(4 + 1))\), but the coefficient part (which is a sub - matrix of the augmented matrix) is square.

So, the set of data that is likely to be represented with a square matrix is when a teacher wants to write an augmented matrix representing a linear system with four equations and four unknowns.

Answer:

A teacher wants to write an augmented matrix representing a linear system with four equations and four unknowns.