1.4 Matrices

To multiply A by B, the number of columns of A must equal the number of rows of B. Each entry of AB is the dot product of a row of A with a column of B.

The order matters: AB may exist while BA does not, or both may exist with different results. Matrix addition requires equal dimensions, while scalar multiplication scales every entry.

For A=[[1,2],[3,4]] and B=[[0,1],[2,0]], AB=[[4,1],[8,3]], whereas BA=[[3,4],[1,2]].

Matrix multiplication is not entry-by-entry multiplication, and AB=BA is a special property rather than a default.

For a 2×2 matrix [[a,b],[c,d]], det A=ad−bc. A square matrix has an inverse only when its determinant is non-zero; determinant magnitude gives the scale factor for oriented area or volume.

A zero determinant means the transformation collapses dimension and distinct vectors become dependent. For larger matrices, expand by a row or column or use row operations while tracking determinant changes.

For [[2,1],[3,2]], det=4−3=1, so the matrix is invertible and preserves area magnitude. For [[1,2],[2,4]], det=0, so no inverse exists.

A negative determinant does not mean “no inverse”; it indicates orientation reversal, while only zero prevents invertibility.

If A is invertible, AA⁻¹=A⁻¹A=I. To solve AX=B, multiply on the left by A⁻¹: X=A⁻¹B. The side matters because matrix multiplication is not generally commutative.

For a product, (AB)⁻¹=B⁻¹A⁻¹: the operations are undone in reverse order. Check dimensions and multiply back to verify the result.

If a transformation first applies B and then A, its matrix is AB; reversing the process uses B⁻¹A⁻¹, not A⁻¹B⁻¹.

Do not divide by a matrix as if it were a scalar, and do not move A⁻¹ across B without changing the order.

A 2×2 matrix maps a vector (x,y) to a new vector by combining its components. Its columns are the images of the standard basis vectors, so the matrix describes rotation, reflection, stretch, shear or a combination.

Composition is matrix multiplication, and the order records which transformation happens first. Determinant describes area scaling; eigenvectors, when present, keep their direction under the transformation.

The matrix [[0,−1],[1,0]] sends (1,0) to (0,1) and (0,1) to (−1,0), a 90° anticlockwise rotation.

Read the coordinate convention carefully: multiplying on the left by a column-vector matrix is not interchangeable with using row vectors.

An invariant point maps to itself. An invariant line maps every point on the line to a point on the same line, even if individual points move along it.

For a matrix transformation, solve T(v)=v for invariant vectors or use the eigenvalue 1 condition. For an invariant line, check that its direction is mapped to a parallel direction and that the line’s location is preserved.

A reflection leaves every point on its mirror line invariant; a rotation about the origin leaves the origin fixed but usually has no other invariant points.

A line being drawn in the same place is not enough: points must map within it, and “invariant” does not mean every point is fixed.