\boxed{\begin{pmatrix} 1 & 3 \\ 3 & 3 \end{pmatrix}}

["Understanding the Matrix \begin{pmatrix} 1 & 3 \ 3 & 3 \end{pmatrix}: Key Properties and Applications", "The matrix \boxed{\begin{pmatrix} 1 & 3 \ 3 & 3 \end{pmatrix}} is a simple yet compelling example in linear algebra with applications across mathematics, engineering, physics, and data science. This symmetric matrix offers insight into fundamental concepts like matrix rank, eigenvalues, determinant, and more, making it a valuable topic for students and professionals alike.", "---", "### What Is the Matrix \begin{pmatrix} 1 & 3 \ 3 & 3 \end{pmatrix}?", "The matrix in question is a ( 2 \ imes 2 ) real-valued matrix defined as:", "[\nA = \begin{pmatrix} 1 & 3 \ 3 & 3 \end{pmatrix}\n]", "It contains positive entries and exhibits symmetry — the entry at row (i), column (j) equals the entry at row (j), column (i) — which simplifies many calculations.", "---", "### Key Matrix Properties", "- Size: ( 2 \ imes 2 ) square matrix\n- Type: Symmetric, real, non-zero\n- Elements:\n - ( a_{11} = 1 )\n - ( a_{12} = a_{21} = 3 ) (off-diagonal entries)\n - ( a_{22} = 3 )", "---", "### Rank and Determinant", "#### Rank\nTo determine the rank (the dimension of the vector space spanned by the matrix’s rows or columns), we compute its determinant:", "[\n\det(A) = (1)(3) - (3)(3) = 3 - 9 = -6\n]", "Since the determinant is non-zero (( \det(A) <br/>\neq 0 )), the matrix is invertible and has full rank — rank 2. This means the two rows (or columns) are linearly independent.", "#### Determinant Value\n[\n\det(A) = -6\n]", "---", "### Eigenvalues and Diagonalization", "To find eigenvalues, solve the characteristic equation:", "[\n\det(A - \lambda I) = 0\n]", "[\nA - \lambda I = \begin{pmatrix} 1 - \lambda & 3 \ 3 & 3 - \lambda \end{pmatrix}\n]", "[\n\det(A - \lambda I) = (1 - \lambda)(3 - \lambda) - 9 = \lambda^2 - 4\lambda - 6\n]", "Solving\n[\n\lambda^2 - 4\lambda - 6 = 0\n]", "using the quadratic formula:", "[\n\lambda = \frac{4 \pm \sqrt{16 + 24}}{2} = \frac{4 \pm \sqrt{40}}{2} = \frac{4 \pm 2\sqrt{10}}{2} = 2 \pm \sqrt{10}\n]", "So the eigenvalues are:", "[\n\lambda_1 = 2 + \sqrt{10},\quad \lambda_2 = 2 - \sqrt{10}\n]", "Both eigenvalues are real and distinct, confirming the matrix is diagonalizable. The eigenvectors can be computed from ( (A - \lambda I)\mathbf{v} = 0 ), supporting applications in systems of differential equations and principal component analysis.", "---", "### Applications of the Matrix", "#### 1. Linear Systems and Transformations\nThis matrix represents a linear transformation in ( \mathbb{R}^2 ) that stretches and shears vectors, useful for modeling constraints or flow dynamics.", "#### 2. Graph Theory\nMatrices like this often appear in adjacency-like structures for graphs, helping analyze connectivity or network behavior.", "#### 3. Data Analysis and Machine Learning\nIn datasets with correlated features, symmetric matrices like this help with covariance, PCA (Principal Component Analysis), or regularization.", "#### 4. Physics and Mechanics\nUsed in moment of inertia tensors and stress-strain models, where off-diagonal terms capture coupling effects.", "---", "### Why Study This Matrix?", "Although simple, \boxed{\begin{pmatrix} 1 & 3 \ 3 & 3 \end{pmatrix}} embodies core linear algebra principles — full rank, determinant, eigenvalues — enabling understanding of more complex matrices. Its symmetric form ensures real eigenvalues and orthogonal eigenvectors, foundational for quadratic forms and optimization.", "---", "### Summary", "| Property | Value/Description |\n|------------------|---------------------------------------|\n| Dimensions | ( 2 \ imes 2 ) |\n| Determinant | ( -6 ) |\n| Rank | 2 (Full rank, invertible) |\n| Eigenvalues | ( 2 \pm \sqrt{10} ) |\n| Symmetric | Yes |\n| Applications | Linear algebra, graphs, physics, ML |", "---", "### Key Takeaway", "Studying matrices like\n[\n\begin{pmatrix} 1 & 3 \ 3 & 3 \end{pmatrix}\n] deepens understanding of linear transformations, eigenvalues, and rank — pivotal concepts underpinning modern applied mathematics and data science.", "---", "Keywords: matrix properties, 2x2 matrix, determinant, eigenvalues, symmetric matrix, linear algebra, applied math, PCA, graph theory, inverse matrix", "---", "References & Further Reading\n- Axler, S. Linear Algebra Done Right — for eigenvalues and determinants\n- Strang, G. Introduction to Linear Algebra — for applications and matrix theory\n- Khan Academy – Linear Algebra module — interactive explanations", "---", "Explore matrices like \begin{pmatrix} 1 & 3 \ 3 & 3 \end{pmatrix} to unlock stronger foundations in mathematics and technologies built upon it!"]