\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 11 \\ 15 \end{pmatrix}

["Understanding Matrix Multiplication: Solving (\begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 11 \ 15 \end{pmatrix})", "Matrix multiplication is a fundamental concept in linear algebra, widely used in mathematics, computer science, engineering, and data science. One common marine problem you might encounter involves solving systems of equations represented visually through matrices. In this article, we’ll explore how to solve the matrix equation (\begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 11 \ 15 \end{pmatrix}) step by step, uncover the values of unknowns, and discuss the broader implications of matrix operations.", "---", "### What is Matrix Multiplication?", "Matrix multiplication is not element-wise; it involves taking the dot product of rows and columns. Given a (2 \ imes 2) matrix:", "[\n\begin{pmatrix} a & b \ c & d \end{pmatrix}\n]", "and a column vector:", "[\n\begin{pmatrix} 2 \ 3 \end{pmatrix},\n]", "the result of the multiplication is another (2 \ imes 1) matrix (a column vector):", "[\n\begin{pmatrix} a\cdot 2 + b\cdot 3 \ c\cdot 2 + d\cdot 3 \end{pmatrix} = \begin{pmatrix} 11 \ 15 \end{pmatrix}.\n]", "This setup defines a system of two linear equations:", "[\n\begin{cases}\n2a + 3b = 11 \quad &\ ext{(Equation 1)} \\n2c + 3d = 15 \quad &\ ext{(Equation 2)}\n\end{cases}\n]", "---", "### Step 1: Solving for (a) and (b)", "From Equation 1:", "[\n2a + 3b = 11\n]", "To eliminate one variable, express (a) in terms of (b) (or vice versa). For example, solving for (a):", "[\n2a = 11 - 3b \quad \Rightarrow \quad a = \frac{11 - 3b}{2}\n]", "Since (a) and (b) represent unknowns in a general system, infinitely many solutions exist unless additional constraints are provided. However, matrix equations usually require unique solutions through full linear systems. Since we only have two equations and four variables, this indicates a subset of constraints — meaning (b) (or (c), (d)) can be freely chosen to represent families of solutions.", "Let’s pick a simple value: suppose (b = 1). Substituting:", "[\na = \frac{11 - 3(1)}{2} = \frac{8}{2} = 4\n]", "Check: (2(4) + 3(1) = 8 + 3 = 11) ✅", "Thus, one possible pair is:", "[\na = 4, \quad b = 1\n]", "---", "### Step 2: Solving for (c) and (d)", "Now use Equation 2:", "[\n2c + 3d = 15\n]", "Similarly, solve for (c):", "[\n2c = 15 - 3d \quad \Rightarrow \quad c = \frac{15 - 3d}{2}\n]", "Setting (d = 1) (again arbitrary), we get:", "[\nc = \frac{15 - 3(1)}{2} = \frac{12}{2} = 6\n]", "Verification: (2(6) + 3(1) = 12 + 3 = 15) ✅", "---", "### The Full Matrix Solution", "With (b = 1), (d = 1), we obtain:", "[\na = 4, \quad b = 1 \\nc = 6, \quad d = 1\n]", "Thus, the matrix becomes:", "[\n\begin{pmatrix} 4 & 1 \ 6 & 1 \end{pmatrix}\n]", "Multiplying:", "[\n\begin{pmatrix} 4 & 1 \ 6 & 1 \end{pmatrix} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 8 + 3 \ 12 + 3 \end{pmatrix} = \begin{pmatrix} 11 \ 15 \end{pmatrix}\n]", "Our solution satisfies the equation exactly.", "---", "### Why This Matters: Applications in Real Life", "Matrix equations like this apply in multiple domains:", "- Computer Graphics: Transformations of objects using homography matrices.\n- Linear Systems: Solving multiple equations simultaneously in engineering.\n- Machine Learning: Weight updates in neural networks use matrix representations.\n- Economics: Modeling input-output relationships in production systems.", "Each cell and row in a matrix encodes meaningful data relationships — understanding how to operate on them empowers deeper problem-solving.", "---", "### Summary", "Solving\n[\n\begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 11 \ 15 \end{pmatrix}\n]\nreduces to solving two linear equations:\n[\n2a + 3b = 11 \quad \ ext{and} \quad 2c + 3d = 15.\n]\nWithout additional constraints, multiple solutions exist—demonstrating the richness of linear systems. A valid pair of parameters is (a = 4), (b = 1), (c = 6), (d = 1), giving the matrix:\n[\n\boxed{\n\begin{pmatrix} 4 & 1 \ 6 & 1 \end{pmatrix}\n}\n]\nwhich accurately transforms the input vector (\begin{pmatrix} 2 \ 3 \end{pmatrix}) into (\begin{pmatrix} 11 \ 15 \end{pmatrix}).", "---", "### Frequently Asked Questions (FAQ)", "Q: Can a matrix equation always be solved uniquely?\nA: Not always. A solution is unique only if the number of independent equations matches the number of unknowns. Here, two equations for four variables imply infinitely many solutions — a typical scenario in real-world linear systems.", "Q: How do we find all possible solutions?\nA: Express variables in terms of free parameters. For example, let (b) and (d) be free variables, then:\n[\na = \frac{11 - 3b}{2}, \quad c = \frac{15 - 3d}{2}\n]", "Q: Are matrices useful for systems without unique solutions?\nA: Yes — they model the entire set of feasible solutions and help analyze system behavior, stability, and transformations.", "---", "Understanding how to multiply matrices and solve corresponding linear systems unlocks powerful tools for modeling complex relationships across disciplines. Keep practicing with different values and vector inputs to strengthen your grasp of linear algebra fundamentals.", "---", "Keywords: matrix multiplication, linear equations, (2 \ imes 2 matrix*, solving linear systems, linear algebra tutorial, matrix transformation, vector multiplication, system of equations, computational math."]