If A is a 5×3 matrix, then the matrix-vector product A
→
x
is defined for any vector
→
x
∈Rn, where

Question

If A is a 5×3 matrix, then the matrix-vector product A
→
x
 is defined for any vector 
→
x
∈Rn, where

BlackTom AI · Accepted Answer

When considering a matrix-vector product A x, the inner dimension must match between the matrix and the vector.
Option 1: n = 1. This would mean x is a scalar, but multiplying a 5×3 matrix by a 1-dimensional vector is not defined, since the inner dimensions 3 and 1 do not match. So this is incorrect.
Option 2: n = 3. This aligns with A being 5×3, where the vector x must be in R^3 for the product A x to be defined. The multiplication essentially collapses the 3 columns of A with the 3 entries of x, resulting in a 5-dimensional vector. This is the correct interpretation.
Option 3: n = 5. If x were in R^5, the inner dimensions would be 3 and 5, which do not match, making the product undefined. Therefore this choice is incorrect.
Option 4: n could be any positive integer. In reality, n is fixed by the number of columns of A, which is 3 for a 5×3 matrix. Saying any positive integer would ignore the dimensional compatibility requirement, so this is incorrect.

MAT223H1S Winter 2025 Reading Quiz 2

If A is a 5×3 matrix, then the matrix-vector product A → x is defined for any vector → x ∈Rn, where

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