Understanding the Problem

We are given the product of two matrices. The first matrix is a 2x2 matrix, and the second is a 2x1 matrix. The product of an (m x n) matrix and an (n x p) matrix results in an (m x p) matrix. Here, (2x2) * (2x1) will yield a (2x1) matrix.

Step-by-Step Solution

Step 1: Identify the Matrices

Let Matrix A be the first 2x2 matrix: $A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

Let Matrix B be the second 2x1 matrix: $B = [\begin{matrix} 5 \\ 6 \end{matrix}]$

Step 2: Set Up the Multiplication

The product C = A * B is calculated as follows: $[\begin{matrix} c_{11} \\ c_{21} \end{matrix}] = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} 5 \\ 6 \end{matrix}]$

Step 3: Calculate Each Element of the Result

Element c₁₁ is the dot product of the first row of A and the first (and only) column of B: $c_{11} = (1 \times 5) + (2 \times 6) = 5 + 12 = 17$

Element c₂₁ is the dot product of the second row of A and the first (and only) column of B: $c_{21} = (3 \times 5) + (4 \times 6) = 15 + 24 = 39$

Step 4: Write the Final Answer

The resulting product matrix is: $[\begin{matrix} 17 \\ 39 \end{matrix}]$

Comparing this to the provided options, the correct answer is the first option: $[\begin{matrix} 17 \\ 39 \end{matrix}]$

Related Topics & Formulae

Matrix Multiplication: The product of two matrices A (of size m x n) and B (of size n x p) is a new matrix C (of size m x p). Each element c_ij is computed as the dot product of the i-th row of matrix A and the j-th column of matrix B.

Formula: $c_{i j} = \sum_{k = 1}^{n} a_{i k} b_{k j}$

Key Theory: For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix has the number of rows from the first matrix and the number of columns from the second matrix.