DPST1013 Chap.5 Matrices, Inverses and Determinants

Matrix multiplication represents composition. Applying B and then A produces AB, while reversing the order applies transformations differently and may not even be dimensionally defined. Products generally cannot be commuted like ordinary numbers.

The transformed columns are dependent and the transformation collapses dimension, so different inputs can share the same output. No inverse can recover every input, and an associated system needs a separate consistency analysis.

A row swap reverses its sign, scaling one row scales the determinant by the same factor, and adding a multiple of another row leaves it unchanged. Track these effects during elimination.

For a square matrix, any one of several equivalent tests is enough: a non-zero determinant, a pivot in every row and column, linearly independent columns, or reduction to the identity. Choose the test suited to the available work. A non-square matrix has no two-sided inverse in this setting. If a determinant vanishes, stop the inverse algorithm and explain the resulting loss of information.

The product AB applies B first and then A. To undo the composition, apply the inverse of A first and the inverse of B second, giving B inverse times A inverse. Multiplying on both sides verifies that the middle factors collapse to identities. Writing A inverse times B inverse usually fails because matrices need not commute and would undo the transformations in the wrong sequence.

Expand along a row or column containing many zeros, since every zero removes a minor calculation. Track the alternating sign pattern carefully and retain exact arithmetic. For a dense large matrix, elimination to triangular form is often faster, provided every row swap and scaling is reflected in the determinant. The best method depends on structure; there is no reward for using the longest universal procedure.

The absolute determinant is the scale factor for oriented area in two dimensions or volume in three dimensions. A negative determinant also reverses orientation, while a zero determinant collapses the region into lower dimension. This interpretation predicts whether a numerical answer is plausible. For example, a determinant of magnitude two must double area even if individual coordinates change in less obvious ways.

If multiple right-hand sides share the same invertible coefficient matrix A, computing A inverse once allows each solution to be written as A inverse times the relevant vector. For a single numerical system, row reduction may be quicker and more stable.

Whichever route you use, multiply the coefficient matrix by the proposed solution to reproduce the original right-hand side rather than checking only intermediate matrix arithmetic.