Matrices
A system of equations is mostly repeated letters around a block of numbers. What can be done with the block alone?
Where you are 12 inside
What this area is about
A matrix is a rectangle of numbers with its own arithmetic. It packs a whole linear system into one equation AX = B, and for a square A one number, the determinant, says whether the system has exactly one solution.
How these fit together
Definitions. Rows, columns and dimensions names sizes and entries; the transpose swaps rows with columns.
Arithmetic. Adding and scaling works entry by entry on matrices of one size. Matrix multiplication pairs rows with columns, so AB needs as many columns in A as rows in B. Writing a system as a matrix equation puts it to work.
Determinants. Computing a determinant and cofactor expansion are procedures; what the determinant means (area or volume scale) and its properties (row operations, \det(AB) = \det A \det B) are results.
Solving. A square A is invertible exactly when \det A \ne 0. The 2 by 2 inverse, row reducing A next to I and Cramer’s rule then solve AX = B.
A path through it
Vocabulary first. Then adding, multiplying, and the transpose or the matrix equation. Separately, compute a determinant. The 2 \times 2 inverse needs both strands; Cramer’s rule and the row-reduction inverse follow it, the second also needing row reduction. The determinant’s meaning (which needs linear transformations) comes before cofactor expansion, then properties, which need row operations.
Next: matrices as transformations and row reduction.