The matrix-vector product

(1 2 1
−1 0 1)(1
2
3)

is a vector in Rn, where

Question

The matrix-vector product

(1	2	1
−1	0	1)(1
2
3)

is a vector in Rn, where

BlackTom AI · Accepted Answer

Consider the given matrix and vector and examine their dimensions to determine where the product lives.
Option 1: n=1. This would place the result in R^1, but a 2×3 matrix multiplied by a 3×1 vector yields a 2×1 vector, not a scalar. So this is incorrect.
Option 2: n=2. Here the result is a vector with two components, i.e., in R^2, which matches the product of a 2×3 matrix with a 3×1 vector. This aligns with the standard rules for matrix-vector multiplication, where the output dimension equals the number of rows of the matrix. This option is consistent with the computation.
Option 3: n=3. This would imply a 3-dimensional output, but the 2×3 matrix cannot produce a 3-dimensional result from a 3×1 vector; the output dimension is determined by the number of rows, which is 2. So this is incorrect.
Option 4: This matrix-vector product is undefined. Given the matrix is 2×3 and the vector is 3×1, the inner dimensions (3) match, making the product defined. Therefore, this statement is false.
In summary, dimensions dictate that the product yields a 2-component vector, confirming n=2 as the correct choice and showing why the other options do not fit the dimensional results or the definition of matrix-vector multiplication.

MAT223H1S Winter 2025 Reading Quiz 2

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

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