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Download (Some) Matrices and Determinants
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(Some) Matrices and Determinants Note: In the matrices below, lower case letters may be either numbers or variable expressions. • For a 2x2 matrix A, the determinate ∆ is calculated by: • ∆ For a 3x3 matrix A, the determinant ∆ is calculated by: | | | ∆ The second representation of the determinant matrix is just to show where the math comes from. Procedure: 1. Repeat the first two columns (after red dashed line) 2. Multiply diagonals. From left to right is a positive contribution, from right to left is a negative contribution (note that signs may be changed by actual matrix components when multiplication performed). Cramer’s Rule (aka some Dinosaur Math) Given three matrices A (nxn), X (nx1) and Y (nx1), where n is an integer, , You can solve for unknowns X1, X2, …, Xn by solving for a series of determinants and taking appropriate ratios. As above, ∆ = |A| For the remainder of the required determinants, i=1…n, replace the ith column in the original A matrix with the Y matrix as follows: 1 3 2 ∆ . . . . . . . ∆ All the way through . . . . ∆ Then: ∆ ∆ , ∆ ∆ ,… ∆ ∆ 1 1