Vectors and Matrices
Core ideas
Vector notation in two and three dimensions, magnitude, unit vectors, scalar (dot) product, angle between vectors, and perpendicularity. Matrices: addition, multiplication, determinants, inverses, solving linear systems, and transformations including rotations and reflections. Understanding when a matrix is singular and the meaning of the determinant as an area/volume scale factor is essential.
Traps
- Matrix multiplication is not commutative: AB ≠ BA in general.
- The dot product is a scalar, not a vector; the cross product (if studied) is a vector.
- A 2×2 inverse requires determinant ad - bc ≠ 0; forgetting to divide by the determinant is a common error.
- When solving AX = B, pre-multiply by A^{-1}: X = A^{-1}B, not BA^{-1}.
- Check dimensions before multiplying matrices.
Rule of thumb
For a 2×2 matrix [[a,b],[c,d]], the inverse is (1/(ad-bc))[[d,-b],[-c,a]]. Verify by multiplying to get the identity.
Worked example
Solve the system 2x + y = 5, x + 3y = 10 using matrices. A = [[2,1],[1,3]], det = 6 - 1 = 5, so A^{-1} = (1/5)[[3,-1],[-1,2]]. Then X = A^{-1}B = (1/5)[[3,-1],[-1,2]][[5],[10]] = (1/5)[[15-10],[-5+20]] = (1/5)[[5],[15]] = [[1],[3]]. So x = 1, y = 3; check: 2(1)+3 = 5 and 1+9 = 10.