Questions: Matrices: Definition, Notation, and Special Types

Question 1

In a matrix A, which entry does the notation a₃₂ refer to?

• A. The entry in column 3, row 2
• B. The entry in row 3, column 2
• C. The entry in row 2, column 3
• D. The entry at position 32 counted left-to-right, top-to-bottom

Answer: (B)

In the notation a_ij, the first subscript i always indicates the row and the second subscript j always indicates the column. So a₃₂ is the entry in row 3, column 2. This convention — row first, then column — is consistent throughout all of linear algebra, regardless of the operation. The confusion between a₃₂ and a₂₃ is extremely common and can cause significant errors in matrix multiplication and other operations.

Question 2

You are given a square matrix A where a_ij = 1 if i = j and a_ij = 0 if i ≠ j. What type of special matrix is this?

• A. A symmetric matrix, because it equals its own transpose
• B. The identity matrix, because it has 1s on the main diagonal and 0s elsewhere
• C. A diagonal matrix, but not the identity matrix
• D. An upper triangular matrix with 1s on the diagonal

Answer: (B)

This definition — 1s exactly where the row index equals the column index (the main diagonal), and 0s everywhere else — is precisely the identity matrix I. While it is also symmetric and diagonal, the most specific and important classification is the identity matrix. It behaves like the number 1 in matrix multiplication: for any compatible matrix A, IA = AI = A. Recognizing it from its definition (rather than just memorizing 'it's the identity') is essential for understanding why it has this property.

Question 3

A 3 × 5 matrix has more columns than rows.

• T. True
• F. False

Answer: True

Matrix dimensions are always stated as m × n where m is the number of rows and n is the number of columns. A 3 × 5 matrix has 3 rows and 5 columns — more columns than rows. This is a non-square matrix. Getting the convention right matters immediately: a 3 × 5 and a 5 × 3 matrix are completely different objects (the first has 3 rows and 5 columns; the second has 5 rows and 3 columns), and matrix multiplication rules depend critically on which dimension is which.

Question 4

In the notation a_ij for a matrix entry, the subscript j refers to the row number.

• T. True
• F. False

Answer: False

In a_ij, i always refers to the row and j always refers to the column — this convention never changes. The subscript j is the column index. Reversing this is one of the most common mistakes in early linear algebra, and it propagates: an error in reading a_ij causes errors in matrix multiplication, system setup, and interpreting results. The row-then-column order in a_ij mirrors the m × n dimension notation, which also states rows first.

Question 5

Why does the order of subscripts in a_ij matter, and what error arises if a student consistently reads j as the row index and i as the column index?

Model answer: The order matters because a_ij and a_ji refer to different entries — unless the matrix is symmetric, these are generally different values. If a student swaps the convention, they are effectively reading the transpose of the matrix instead of the matrix itself. Every computation that depends on specific entries — matrix multiplication, solving systems, checking symmetry — will produce wrong results.

The subscript order a_ij (row i, column j) is a bookkeeping convention, but it is the universal convention, and violating it silently introduces errors that can be hard to trace. When two matrices are multiplied, the (i,j) entry of the product is the dot product of row i of the first matrix and column j of the second — if you have the indices swapped, you will select the wrong rows and columns. Fluency with this notation is genuinely foundational to everything that follows in linear algebra.