The Multiplication of Row Vector and Column Vector: Explaining the 1 × 1 Matrix Result

If defined, a row times a column is always a 1 × 1 matrix.

When defining the multiplication of a row vector and a column vector, the resulting matrix is always a 1 × 1 matrix, also known as a scalar

When defining the multiplication of a row vector and a column vector, the resulting matrix is always a 1 × 1 matrix, also known as a scalar.

To understand why this is the case, let’s consider the dimensions of the row and column vectors involved.

A row vector has dimensions 1 × n, where n represents the number of elements in the row vector. Similarly, a column vector has dimensions m × 1, where m represents the number of elements in the column vector.

To perform the multiplication of a row vector by a column vector, we multiply each element of the row vector by the corresponding element in the column vector and then sum up the products. Since both vectors have the same number of elements (either 1 × n and m × 1), the product of each corresponding element is a 1 × 1 matrix.

For example, let’s take a row vector R = [1 2 3] and a column vector C = [4 5 6]. The multiplication of R and C would be:

R × C = [1 2 3] × [4 5 6] = (1*4) + (2*5) + (3*6) = 4 + 10 + 18 = 32.

Here, the result 32 is a 1 × 1 matrix or simply a scalar value. Therefore, it can be said that a row times a column is always a 1 × 1 matrix if defined.

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