What you can now do.

• §8.1 A matrix moves the whole plane. Graph paper that stretches, turns, slants or flips, but never bends. Two landmarks decide everything: 10 lands on column 1 and 01 on column 2. A turn by θ has columns cos θsin θ and −sin θcos θ.
• §8.1 Challenge: build a mirror. The mirror across the y-axis sends 10 to −10 and keeps 01. Same size, turned over: det = −1. Two mirrors give back I.
• §5.1 The determinant as area. The photocopier picture: every area is multiplied by the same number, det A. Read it off the unit square, which lands on the parallelogram of the columns. Negative means flipped, zero means squashed flat and no inverse. In 3D it is the volume of the box made by the columns.
• §5.1 The three rules. det I = 1, a row swap flips the sign, and det is linear in each row (one row at a time). From them: two equal rows give 0, and elimination steps never change det.
• §5.1 From the pivots. Push the leaning stack upright: eliminate to U, multiply the pivots, and add one minus sign per row swap. Also det AB = det A · det B, det A⁻¹ = 1 / det A and det Aᵀ = det A.
• §5.2 Cofactors. Cross out an entry's row and column to get its minor, attach the checkerboard sign + − +, and add entry × cofactor along any row or column. Pick the row with the most zeros.
• §5.3 Cramer's rule and the inverse. The café prices one at a time: xⱼ = det Bⱼ / det A, with b in column j. The whole inverse is A⁻¹ = Cᵀ / det A. Neat on paper, slow for big systems.
• §5.3 Areas, volumes and the cross product. A triangle's area is ½ |det| of two edge vectors. A box's volume is |det| of its three edges. u × v stands at a right angle to u and v, and its length is their parallelogram's area. Torque is τ = r × F.

Next lesson: eigenvalues and eigenvectors (§6.1–6.5)

Most vectors change direction when a matrix moves the plane. A few special ones only get longer or shorter. Lesson 7 finds them, and the determinant is the tool that does it.

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