5السؤال: أوجد المصفوفة $\mathbf{M}$ بحيث أن $\mathbf{M} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 11 \\ 15 \end{pmatrix}$.

["How to Find Matrix $\mathbf{M}$ That Transforms $\begin{pmatrix} 2 \ 3 \end{pmatrix}$ to $\begin{pmatrix} 11 \ 15 \end{pmatrix}$: A Step-by-Step Guide", "When faced with a linear equation involving matrices, such as\n$$\n\mathbf{M} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 11 \ 15 \end{pmatrix},\n$$\nthe challenge is to determine the unknown $2 \ imes 2$ matrix $\mathbf{M}$. This type of problem arises frequently in linear algebra and has practical applications in transformations, computer graphics, and system modeling. In this article, we’ll walk through how to find $\mathbf{M}$ step by step.", "---", "### What Is Matrix Multiplication?", "Let\n$$\n\mathbf{M} = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \quad \ ext{where } a, b, c, d \ ext{ are unknowns}.\n$$", "Then the multiplication\n$$\n\mathbf{M} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 2a + 3b \ 2c + 3d \end{pmatrix}\n$$\nmust equal\n$$\n\begin{pmatrix} 11 \ 15 \end{pmatrix}.\n$$", "This gives a system of two equations:\n$$\n2a + 3b = 11 \quad \ ext{(1)}\n$$\n$$\n2c + 3d = 15 \quad \ ext{(2)}\n$$", "---", "### Analyzing the System", "Each row of $\mathbf{M}$ produces one component of the output vector. Since we have only one vector input, the solution depends on choosing values for the entries $a, b$ and $c, d$ that satisfy both equations simultaneously.", "Because there are three unknowns per equation but only two equations, the system is underdetermined—meaning infinitely many matrices $\mathbf{M}$ satisfy the equation. However, we can find a particular solution and describe the general form.", "From equation (1):\n$$\n2a + 3b = 11\n\Rightarrow a = \frac{11 - 3b}{2}\n$$", "From equation (2):\n$$\n2c + 3d = 15\n\Rightarrow c = \frac{15 - 3d}{2}\n$$", "---", "### Constructing a Particular Solution", "To illustrate, pick simple values for one variable. Let’s set $b = 1$.\nThen from (1):\n$$\n2a + 3(1) = 11 \Rightarrow 2a = 8 \Rightarrow a = 4\n$$", "Now set $d = 3$. Then from (2):\n$$\n2c + 3(3) = 15 \Rightarrow 2c + 9 = 15 \Rightarrow 2c = 6 \Rightarrow c = 3\n$$", "Thus, one possible matrix is:\n$$\n\mathbf{M} = \begin{pmatrix} 4 & 1 \ 3 & 3 \end{pmatrix}\n$$", "Let’s verify:\n$$\n\begin{pmatrix} 4 & 1 \ 3 & 3 \end{pmatrix} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 4\cdot2 + 1\cdot3 = 8 + 3 = 11 \ 3\cdot2 + 3\cdot3 = 6 + 9 = 15 \end{pmatrix}\n$$\n✅ Correct.", "---", "### General Solution: All Possible Matrices $\mathbf{M}$", "Since there are two independent equations and four unknowns, the solution space has dimension $4 - 2 = 2$, meaning a 2-dimensional affine space of solutions.", "We can express $\mathbf{M}$ in terms of two free parameters. For example, from above:", "Let $b = t$, $d = s$, then:\n$$\na = \frac{11 - 3t}{2}, \quad c = \frac{15 - 3s}{2}\n$$", "The general solution is:\n$$\n\mathbf{M} = \begin{pmatrix} \frac{11 - 3t}{2} & t \ \frac{15 - 3s}{2} & s \end{pmatrix}, \quad t, s \in \mathbb{R}\n$$", "---", "### Applications and Why This Matters", "Understanding how to determine unknown matrices via linear transformations is valuable in:", "- Computer graphics: Defining how objects scale, rotate, or translate.\n- Data science: Constructing weight matrices in transformations.\n- Engineering: Solving systems where relationships between inputs and outputs are linear.", "---", "### Conclusion", "To find the matrix $\mathbf{M}$ such that $\mathbf{M} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 11 \ 15 \end{pmatrix}$, express $\mathbf{M}$ as\n$$\n\begin{pmatrix} a & b \ c & d \end{pmatrix},\n$$\nthen solve the linear system $2a + 3b = 11$ and $2c + 3d = 15$.\nWith infinitely many solutions, the general form includes free parameters:\n$$\n\mathbf{M} = \begin{pmatrix} \frac{11 - 3t}{2} & t \ \frac{15 - 3s}{2} & s \end{pmatrix}, \quad t,s \in \mathbb{R}\n$$", "This method enables precise control over linear transforms—essential in science and technology.", "---", "Keywords:\n$\mathbf{M} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 11 \ 15 \end{pmatrix}$, matrix equation, linear algebra, solve matrix, vector transformation, 2x2 matrix, affine solution space.", "---", "Meta description:\nLearn how to find matrix $\mathbf{M}$ such that it transforms $\begin{pmatrix} 2 \ 3 \end{pmatrix}$ into $\begin{pmatrix} 11 \ 15 \end{pmatrix}$. Discover general solutions and applications in science and engineering."]