The matrix-vector product

(1 −1
2 0
1 1)(1
2
3)

is a vector in Rn, where

Question

The matrix-vector product

(1	−1
2	0
1	1)(1
2
3)

is a vector in Rn, where

BlackTom AI · Accepted Answer

Frame the problem by clearly identifying the objects involved: the matrix is presented as a 3-by-2 matrix
[ [1, -1], [2, 0], [1, 1] ] and the vector is a 3-by-1 column [1, 2, 3]^T. The matrix-vector product is defined only when the number of columns of the matrix matches the number of rows of the vector.

Option 1: n=1. This would imply a vector in R^1, but that ignores the actual shapes of the operands. The matrix has 2 columns and the vector has 3 entries, so a 1-dimensional output cannot even be formed from these operands. Therefore this choice is inconsistent with the given shapes.

Option 2: n=2. Here the confusion might be that the matrix has 2 columns, suggesting a 2-dimensional domain, but the vector has 3 entries, which mismatches the required inner dimension for multiplication. As a result, the product is not defined despite any attempt to force an n=2 interpretation.

Option 3: n=3. This would correspond to an output vector in R^3, yet the product is not feasible because the inner dimensions do not align: a 3-by-2 matrix cannot be multiplied by a 3-by-1 vector since the number of columns (2) does not equal the number of rows of the vector (3).

Option 4: This matrix-vector product is undefined. Given the shapes 3×2 and 3×1, the inner dimensions fail to match (2 ≠ 3), so the standard matrix-vector multiplication is not possible. This is the precise, formal reason for the undefined status.

In summary, the only statement that accurately reflects the situation is that the product is undefined due to a mismatch in inner dimensions, while the other dimensional-n arguments misinterpret the shape constraints or overlook the core requirement for matrix-vector multiplication.

MAT223H1S Winter 2025 Reading Quiz 2

The matrix-vector product (1 −1 2 0 1 1)(1 2 3) is a vector in Rn, where

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