Question

which of the following sets of data would you likely represent with a square matrix?
a shoe salesperson wants to use an augmented matrix to represent the quantity of children’s 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 is a matrix with the same number of rows and columns.

Step2: Analyze each option

• Option 1:

For the shoe salesperson, if we consider children's shoes and adult shoes (2 types) and 2 days, an augmented matrix would have \(2\) rows (for 2 days) and \(2 + 1=3\) columns (2 for shoe types and 1 for total or other related quantity). So, it is not a square matrix.

• Option 2:

If there are \(n\) members in the racquetball club, the matrix representing the scores where each member plays every other member will have \(n\) rows (for each member as a row - representing the games they play) and \(n\) columns (for each member as a column - representing the games they receive). But we don't know the number of members. If \(n = 2\), it is a square matrix, but in general, we can't be sure.

• Option 3:

A coefficient matrix for a linear system with \(2\) equations and \(3\) unknowns has \(2\) rows and \(3\) columns. Since \(2
eq3\), it is not a square matrix.

• Option 4:

A linear system with \(4\) equations and \(4\) unknowns. An augmented matrix for a linear system \(Ax = b\) where \(A\) is the coefficient matrix. If \(A\) is a \(4\times4\) matrix (because there are 4 equations and 4 unknowns), the augmented matrix \([A|b]\) will have \(4\) rows (one for each equation) and \(4 + 1=5\) columns. But the coefficient matrix \(A\) is a square matrix.

Answer:

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