Questions: Matrix Addition and Subtraction

Question 1

A student wants to compute A + B where A is a 2×3 matrix and B is a 3×2 matrix. They transpose B to make it 2×3, then add entry-by-entry, reasoning that both now have the same total number of entries. What is wrong with this approach?

• A. Nothing is wrong — addition is defined as long as both matrices have the same number of entries
• B. The student should transpose A instead of B, since convention requires adding along rows
• C. Matrix addition requires identical dimensions; transposing B creates a mathematically different matrix, and the result is not a valid sum of A and the original B
• D. The approach is valid but inefficient — direct addition without transposing would give the same result

Answer: (C)

Matrix addition is defined only for matrices with identical dimensions — same number of rows AND same number of columns. Transposing B changes the matrix: the (i,j) entry of B becomes the (j,i) entry of B^T, a different mathematical object. You cannot add a 2×3 and a 3×2 by reshaping one of them; A + B is simply undefined when their dimensions don't match.

Question 2

Which property does matrix addition NOT share with scalar addition?

• A. Commutativity: A + B = B + A
• B. Associativity: (A + B) + C = A + (B + C)
• C. Existence of an additive identity: there is a matrix 0 such that A + 0 = A
• D. Unrestricted operability: A + B is defined even when A and B have different dimensions

Answer: (D)

Scalar addition works for any two numbers regardless of 'dimension,' but matrix addition requires both matrices to have identical dimensions. Commutativity, associativity, and additive identity all hold for matrix addition and follow entry-by-entry from the same properties of scalar addition. The dimension requirement is an additional constraint with no scalar analogue.

Question 3

Matrix addition proceeds entry-by-entry: the (i,j) entry of A + B equals a_ij + b_ij.

• T. True
• F. False

Answer: True

This is the definition of matrix addition. For each position (i,j), you simply add the scalar entries at that position. This locality is what makes all algebraic properties inherit directly from scalar addition, and it makes matrix addition fundamentally different from matrix multiplication, which mixes entries across rows and columns.

Question 4

Matrix multiplication, like matrix addition, is defined entry-by-entry: (AB)_ij = a_ij · b_ij.

• T. True
• F. False

Answer: False

This is a critical misconception to correct before studying multiplication. Addition IS entry-by-entry, but multiplication is NOT. The (i,j) entry of AB is the dot product of row i of A with column j of B: (AB)_ij = Σ_k a_ik · b_kj. This is why multiplication requires the number of columns of A to equal the number of rows of B, and why multiplication is generally not commutative. Confusing entry-wise with dot-product is one of the most common early errors in linear algebra.

Question 5

Why do m×n matrices under addition form a vector space, and what does this tell us about the relationship between matrices and vectors in ℝ^(mn)?

Model answer: Matrix addition satisfies all vector space axioms entry-by-entry: commutativity, associativity, the zero matrix as additive identity, additive inverses (−A), and scalar multiplication by scaling each entry. These axioms hold because they hold for each scalar entry, and matrix operations apply them uniformly. An m×n matrix can be 'unrolled' into a vector of mn entries — M_{m,n} is isomorphic to ℝ^(mn) as a vector space. Matrices are vectors with entries arranged in a grid.

Recognizing M_{m,n} as a vector space is not just an abstraction — it enables results that treat spaces of matrices as geometric objects (measuring distance between matrices, defining linear maps between them, identifying subspaces). The simplicity of entry-wise addition makes this structure transparent, and it sets up the contrast with matrix multiplication: addition is uniform and commutative; multiplication is structural and generally noncommutative.