$ egin{vmatrix} 9 & -18 \ -18 & 20 \end{vmatrix} = (9)(20) - (-18)^2 = 180 - 324 = -144 $

["# Understanding Determinants: Calculating $ \begin{vmatrix} 9 & -18 \ -18 & 20 \end{vmatrix} = -144$", "When studying linear algebra, one of the key concepts you encounter is the determinant of a 2×2 matrix. The determinant provides important information about a matrix, including whether it is invertible and the area (or volume) scaling factor of the linear transformation it represents.", "In this article, we’ll explore how to compute the determinant of the matrix\n$$\n\begin{bmatrix}\n9 & -18 \\n-18 & 20\n\end{bmatrix}\n$$\nand demonstrate the calculation step-by-step using the formula for 2×2 determinants.", "---", "## What is a Determinant?", "For a 2×2 matrix:\n$$\nA = \begin{bmatrix}\na & b \\nc & d\n\end{bmatrix},\n$$\nthe determinant is calculated as:\n[\n\det(A) = ad - bc\n]\nThis value is foundational in solving systems of linear equations, finding matrix inverses, and analyzing geometric transformations such as rotations, scalings, and reflections.", "---", "## Step-by-Step Calculation", "Given matrix:\n$$\n\begin{bmatrix}\n9 & -18 \\n-18 & 20\n\end{bmatrix}\n$$\nwe identify the values:\n- ( a = 9 )\n- ( b = -18 )\n- ( c = -18 )\n- ( d = 20 )", "Now apply the determinant formula:\n[\n\det = (9)(20) - (-18)(-18)\n]", "Step 1: Multiply the diagonal entries:\n[\n9 \ imes 20 = 180\n]", "Step 2: Multiply the off-diagonal entries (note: both are negative):\n[\n-18 \ imes -18 = 324\n]", "Step 3: Subtract the second product from the first:\n[\n180 - 324 = -144\n]", "So,\n[\n\begin{vmatrix} 9 & -18 \ -18 & 20 \end{vmatrix} = -144\n]", "---", "## Why Determinants Matter", "- Matrix Invertibility: A matrix is invertible only if its determinant is non-zero. Here, $-144 <br/>\neq 0$, so the matrix is invertible.\n- Area Scaling: The absolute value of the determinant represents how much the transformation scales area—here, area would scale by $144$.\n- Eigenvalues and Stability: Determinants feature in characteristic equations that help predict system behavior.", "---", "## Final Thoughts", "Calculating determinants is a quick yet powerful tool in linear algebra. The example $ \begin{vmatrix} 9 & -18 \ -18 & 20 \end{vmatrix} = -144 $ illustrates how basic arithmetic applies directly to this formulaic expression. Whether you're analyzing geometry, solving equations, or studying engineering systems, mastering determinants strengthens your mathematical foundation.", "If you're learning linear algebra or preparing for exams, understanding determinant calculations naturally leads to deeper insights into matrices and vector spaces. Keep practicing — and remember, the determinant is more than just a number: it's a gateway to powerful mathematical concepts.", "---\nKeywords: determinant, 2x2 matrix, linear algebra, linear transformations, matrix calculation, math tutorial, determinant formula", "Meta Description: Learn how to compute the determinant of $ \begin{vmatrix} 9 & -18 \ -18 & 20 \end{vmatrix} $ using $ ad - bc $. This step-by-step guide explains the formula, calculation, and significance in linear algebra."]