$ egin{vmatrix} -6 & 9 \ 4 & -18 \end{vmatrix} = (-6)(-18) - (9)(4) = 108 - 36 = 72 $

["Understanding the Determinant of a 2x2 Matrix: A Step-by-Step Explanation", "When studying linear algebra, one crucial concept is the determinant of a matrix, especially for 2×2 matrices. The determinant provides valuable information about the matrix — such as whether it is invertible and how it transforms areas in geometry. In this article, we dive into the calculation of the determinant for the matrix\n$$\n\begin{vmatrix} -6 & 9 \ 4 & -18 \end{vmatrix}\n$$\nand verify that the result is 72.", "---", "### What Is the Determinant of a 2×2 Matrix?", "For a 2×2 matrix of the form\n$$\nA = \begin{pmatrix} a & b \ c & d \end{pmatrix},\n$$\nthe determinant is computed using the formula:\n$$\n\det(A) = ad - bc\n$$", "This formula stems from the geometric interpretation of the determinant as the signed area of the parallelogram formed by the column vectors of the matrix.", "---", "### Applying the Formula to the Given Matrix", "In our case, the matrix is:\n$$\nA = \begin{pmatrix} -6 & 9 \ 4 & -18 \end{pmatrix}\n$$\nSo we identify:\n- ( a = -6 )\n- ( b = 9 )\n- ( c = 4 )\n- ( d = -18 )", "Now plug into the determinant formula:\n$$\n\det(A) = (-6)(-18) - (9)(4)\n$$", "Calculate each term:\n- ( (-6)(-18) = 108 )\n- ( (9)(4) = 36 )", "Subtract:\n$$\n\det(A) = 108 - 36 = 72\n$$", "---", "### Why Is This Result Important?", "The determinant value of 72 tells us several key things:\n- The matrix is invertible, because 72 ≠ 0.\n- The linear transformation represented by this matrix scales areas by a factor of 72.\n- It also indicates that the two column vectors (-6, 4) and (9, -18) are linearly independent, forming a parallelogram of area 72 in the plane.", "---", "### Final Answer", "$$\n\begin{vmatrix} -6 & 9 \ 4 & -18 \end{vmatrix} = 72\n$$\nThis result demonstrates the straightforward yet powerful method of calculating 2×2 determinants and highlights their significance in linear algebra.", "---", "If you’re studying matrices, remembering the formula ( ad - bc ) and practicing such examples will build a strong foundation for more advanced topics like eigenvalues, matrix inverses, and solving systems of equations.", "Keywords: determinant 2x2 matrix, calculate determinant, linear algebra, matrix calculations, area scaling, inverse matrix, linear transformation.", "---", "📘 Tip: Practice computing determinants with different sets of numbers to master this fundamental concept quickly!"]