What you can now do.

• §2.1 Row and column pictures. The same system read two ways: by rows, lines that cross at the answer; by columns, how much of each column makes b. Parallel lines go with dependent columns.
• §2.2 Elimination. The café puzzle: subtract a multiple of one equation from another so that one unknown disappears. Multiplier = entry ÷ pivot. The crossing point never moves, and the augmented matrix [A | b] keeps just the numbers.
• §2.2 Pivots and back substitution. Clear one column at a time, left to right, until the system is a triangle U. Then the chain of keys: the last row gives the last unknown, which unlocks the row above.
• §2.2 Zero pivots. The relay race: a 0 in a pivot spot means swap in a lower row if one can help. If none can, the matrix is singular: no solution, or infinitely many.
• §2.3 Elimination matrices. Instruction cards: E₂₁ is the identity with −ℓ in row 2, column 1, and E₂₁A does that row step. All the steps together: E₃₂E₃₁E₂₁A = U.
• §2.4 Matrix multiplication. Each entry of AB is a row of A times a column of B. Column j of AB is A times column j of B. A is m by n and B is n by p, so AB is m by p. Order matters: AB is usually not BA.
• §2.6 A = LU. Meal prep: factor once, then solve each new b with two fast triangular solves. L holds the multipliers, each in the spot where it was used. In the lab, lu(A) does not exchange rows: if a zero pivot appears, it stops with the error "zero pivot in row k: this matrix needs a row exchange (PA = LU)". There P is I with its rows swapped: it does the row exchanges first. Lesson 3 covers it.

Next lesson: inverses, transposes and permutations (§2.5, §2.7)

The matrix that undoes A, when it exists and how Gauss-Jordan finds it. Then the transpose, symmetric matrices and the permutation matrices behind row exchanges.

Read along

Professor Gilbert Strang, Introduction to Linear Algebra, 5th edition (2016), sections 2.1 to 2.4 and 2.6. Lectures 1 to 4 of MIT 18.06 on MIT OpenCourseWare cover the same ground.