When multiplying a 3×1 matrix and a 1×3 matrix the resultant matrix is a Group of answer choices 1×1 1×3 3×3 3×1
The correct answer and explanation is:
Correct Answer: 3×3
Explanation:
To understand the result of multiplying a 3×1 matrix and a 1×3 matrix, we must look at the rules of matrix multiplication.
Matrix Dimensions:
Let’s define two matrices:
- Matrix A (3×1): 3 rows and 1 column
Example: A=[a1a2a3]A = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} - Matrix B (1×3): 1 row and 3 columns
Example: B=[b1b2b3]B = \begin{bmatrix} b_1 & b_2 & b_3 \end{bmatrix}
Matrix Multiplication Rule:
Matrix multiplication is defined only when the number of columns in the first matrix equals the number of rows in the second matrix. For these two matrices:
- A has 1 column
- B has 1 row
Since the inner dimensions match (1 = 1), we can multiply them.
Resulting Dimensions:
The resulting matrix has the dimensions of the outer numbers:
- A is 3×1
- B is 1×3
- So the product AB is 3×3
Example:
A=[246],B=[135]A = \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 3 & 5 \end{bmatrix} AB=[2⋅12⋅32⋅54⋅14⋅34⋅56⋅16⋅36⋅5]=[26104122061830]AB = \begin{bmatrix} 2 \cdot 1 & 2 \cdot 3 & 2 \cdot 5 \\ 4 \cdot 1 & 4 \cdot 3 & 4 \cdot 5 \\ 6 \cdot 1 & 6 \cdot 3 & 6 \cdot 5 \end{bmatrix} = \begin{bmatrix} 2 & 6 & 10 \\ 4 & 12 & 20 \\ 6 & 18 & 30 \end{bmatrix}
So, the resulting matrix is 3×3.
This is known as the outer product of two vectors. A column vector (3×1) and a row vector (1×3) produce a square matrix where each element is the product of the respective elements from A and B. This concept is widely used in linear algebra, machine learning, and physics.