Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). One method is to augment the 3×3 matrix with a repetition of the first two columns, giving a 3×5 matrix. We can then express \(x\) and \(y\) as a quotient of two determinants.Įvaluating the Determinant of a 3 × 3 Matrixįinding the determinant of a 2×2 matrix is straightforward, but finding the determinant of a 3×3 matrix is more complicated. The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. \(D_y\):determinant of the numerator in the solution of \(y\). \(D_x\):determinant of the numerator in the solution of \(x\).\(D\):determinant of the coefficient matrix.We can use these formulas to solve for \(x\) and \(y\), but Cramer’s Rule also introduces new notation: Notice that the denominator for both \(x\) and \(y\) is the determinant of the coefficient matrix.
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