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Chapter 3 · Matrices

Exercise 3.1 · Q8 (MCQ): When is A=[aij]m×nA = [a_{ij}]_{m\times n} a square matrix?

Exercise 3.1Types of matrices · Section 3.3
The question

A=[aij]m×nA = [a_{ij}]_{m\times n} is a square matrix, if

(A) m<nm < n
(B) m>nm > n
(C) m=nm = n
(D) None of these

Exam answer

A matrix is a square matrix when its number of rows equals its number of columns.

So A=[aij]m×nA = [a_{ij}]_{m\times n} is square if m=nm = n⁠.

Answer: (C) m=nm = n

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

The subscript m×nm\times n says AA has mm rows and nn columns. A square matrix is, by definition, one whose number of rows equals its number of columns. So AA is square exactly when m=nm = n⁠.

Given, to find, and the plan
Given
A matrix AA of order m×nm\times n⁠: mm rows, nn columns.
To find
The condition on mm and nn for AA to be square.
Plan
(1) Recall the definition of a square matrix. (2) Translate it into a condition on mm and nn⁠. (3) Test each option, with an explicit matrix where an option fails.

Basics you need

1Order m×nm\times n

[aij]m×n[a_{ij}]_{m\times n} denotes a matrix with mm rows and nn columns. Basics: rows, columns and the address

2Square matrix

A matrix is called square when it has as many rows as columns (Section 3.3, type (iii)). Lesson: types of matrices

3Exactly one of m<nm<n⁠, m=nm=n⁠, m>nm>n

For any two numbers, exactly one of these three is true. So the three options (A), (B), (C) never overlap.

Solution

  1. Translate the definition

    AA has mm rows and nn columns. By the definition, AA is square exactly when

    number of rows=number of columns,\text{number of rows} = \text{number of columns},

    that is, when m=nm = n. A square matrix of order n×nn\times n is simply called a square matrix of order nn⁠.

    The definition is the whole argument. It works in both directions: if m=nm = n then rows equal columns, so AA is square; and if AA is square then rows equal columns, so m=nm = n⁠. The name comes from the shape: equal numbers of rows and columns make the array a square block.
    Square matrix: m=nm = n (Section 3.3, type (iii)). Lesson: types of matrices

  2. Examine every option

    For (A) and (B), here are specific matrices satisfying the condition:

    m<n:[123456] (2×3)m>n:[123456] (3×2)\begin{aligned} m < n:&\quad \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\ (2\times3) \\[2mm] m > n:&\quad \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}\ (3\times2) \end{aligned}
    OptionClaimVerdict
    (A)m<nm < nWrong. Fewer rows than columns makes a wide matrix, such as the 2×32\times3 one above: 2 rows but 3 columns, so it is not square.
    (B)m>nm > nWrong. More rows than columns makes a tall matrix, such as the 3×23\times2 one above: 3 rows but 2 columns, so it is not square.
    (C)m=nm = nCorrect. This is exactly the definition (Step 1). For example the 3×33\times3 matrix in Section 3.3 is a square matrix of order 3.
    (D)None of theseWrong, because (C) is correct.
    Since exactly one of m<nm < n⁠, m=nm = n⁠, m>nm > n holds for any order, and only m=nm = n gives a square matrix, (A) and (B) both describe matrices that are never square. One explicit example of each is enough to show the condition does not make AA square.
    One counterexample defeats a claim. Basics: rows and columns
Result

The correct answer is (C): A=[aij]m×nA = [a_{ij}]_{m\times n} is a square matrix if m=nm = n⁠. (Textbook answer: C.)

Check it with numbers

The 2×32\times3 matrix above has 2×3=62\times3 = 6 entries; it cannot be rearranged as a square, since 6 is not a perfect square ✓. A 2×22\times2 matrix (m=n=2m = n = 2⁠) has 4 entries in 2 rows of 2, a square block ✓.

Common mistakes
  1. Mixing up the order: thinking mm is the number of columns. In m×nm\times n⁠, mm counts rows.
  2. Choosing (D) because "square" seems to need extra conditions, for example on the entries. It does not: any entries are allowed, only the shape matters.
What to take away
  1. An m×nm\times n matrix is square exactly when m=nm = n⁠; it is then called square of order nn⁠.
  2. m<nm < n gives a wide matrix and m>nm > n a tall one; neither is square.

Every fact used, and where it comes from

FactUsed inSource
[aij]m×n[a_{ij}]_{m\times n} has mm rows and nn columnsStep 1Section 3.2.1 and its Note (Lesson: what a matrix is)
Square matrix: rows equal columns, m=nm = n⁠; "square matrix of order nn"Steps 1, 2Section 3.3, type (iii) (Lesson: types of matrices)
Orders of the example matricesStep 2Recomputed in the site's verification script
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