Questions: Matrices Introduction

Question 1

A matrix is described as '3 x 5'. What does this tell you about its structure?

• A. 3 columns and 5 rows
• B. 3 rows and 5 columns
• C. 3 diagonals and 5 entries
• D. 5 rows and 3 columns

Answer: (B)

Matrix dimensions are always stated as rows × columns. A 3 × 5 matrix has 3 horizontal rows and 5 vertical columns. This is a frequent source of confusion because people often think of (x, y) coordinates as (columns, rows), which is the opposite convention.

Question 2

Performing a row operation on an augmented matrix — such as multiplying a row by 2 — changes the solution set of the corresponding system of equations.

• T. True
• F. False

Answer: False

Row operations are equivalence transformations: they produce a new matrix whose solution set is identical to the original. Multiplying a row by a nonzero constant, swapping rows, or adding a multiple of one row to another all preserve the solutions. This is the fundamental reason the augmented-matrix method works.

Question 3

What is the purpose of the augmented matrix when solving a system of linear equations?

Model answer: The augmented matrix combines the coefficient matrix and the constant (right-hand side) column into a single rectangular array, so that row operations on the matrix correspond exactly to elimination steps on the original equations.

By appending the constants as an extra column separated by a vertical bar, the augmented matrix captures the full information of the system in compact form. Reducing it to row echelon form is the same as performing systematic elimination, and reading off the solution is equivalent to back-substitution.