Solution: Let $\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Multiply $\mathbf{M}$ with $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ to get:
["# Matrix Multiplication Solved: $\mathbf{M} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ — Complete Result and Explanation", "Matrix multiplication is a fundamental operation in linear algebra with wide applications in computer graphics, machine learning, and engineering. Understanding how two matrices combine empowers deeper insight into transformations and systems of equations. In this article, we focus on solving a clear and common matrix multiplication problem: multiplying a 2×2 matrix $\mathbf{M} = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ with a fixed 2×2 matrix $\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and determining the resulting matrix.", "---", "## The Setup: Matrix Multiplication Steps", "We begin with:", "[\n\mathbf{M} = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \quad A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}\n]", "To compute the product $\mathbf{M} \cdot A$, we use the standard matrix multiplication rule: the element in row $i$ and column $j$ of the resulting matrix is found by taking the dot product of row $i$ from $\mathbf{M}$ with column $j$ from $A$.", "Explicitly:", "[\n\mathbf{M} \cdot A = \begin{pmatrix}\na \cdot 1 + b \cdot 3 & a \cdot 2 + b \cdot 4 \\nc \cdot 1 + d \cdot 3 & c \cdot 2 + d \cdot 4\n\end{pmatrix}\n]", "---", "## The Result: Step-by-Step Calculation", "Putting it all together, we perform each multiplication and addition:", "[\n\begin{aligned}\n\ ext{Top-left entry} &= a \cdot 1 + b \cdot 3 = a + 3b \\n\ ext{Top-right entry} &= a \cdot 2 + b \cdot 4 = 2a + 4b \\n\ ext{Bottom-left entry} &= c \cdot 1 + d \cdot 3 = c + 3d \\n\ ext{Bottom-right entry} &= c \cdot 2 + d \cdot 4 = 2c + 4d\n\end{aligned}\n]", "Thus, the full product is:", "[\n\boxed{\n\mathbf{M} \cdot \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} =\n\begin{pmatrix}\na + 3b & 2a + 4b \\nc + 3d & 2c + 4d\n\end{pmatrix}\n}\n]", "---", "## Why This Matters", "Each entry in the resulting matrix reflects a weighted sum determined by the original components $a, b, c, d$ and the fixed coefficients from matrix $A$. This process exemplifies how matrix multiplication encodes linear combinations — essential for modeling growth rates, transformations, and system dynamics.", "In practical terms, such computations are stepping stones toward:", "- Solving systems of linear equations via matrix inversion\n- Applying linear transformations in computer graphics\n- Conducting efficient data projections in machine learning models", "---", "## Summary", "When multiplying $\mathbf{M} = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ by $\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, the result is:", "[\n\boxed{\n\begin{pmatrix}\na + 3b & 2a + 4b \\nc + 3d & 2c + 4d\n\end{pmatrix}\n}\n]", "This clear computation reinforces core principles of linear algebra and supports advanced mathematical applications across disciplines.", "---", "Keywords: matrix multiplication, 2×2 matrix multiplication, linear algebra, matrix product, vector transformations, mathematical computation, linear transformations, system of equations, computational linear algebra."]
