Matrices and Determinants Cheat Sheet

Key Definitions

• Matrix: A rectangular array of numbers arranged in rows and columns. ✅

• Determinant: A scalar value that is a function of a square matrix. ✅

• Transpose: The matrix obtained by swapping rows and columns. ✅

Important Concepts

Matrix Operations

• Addition: Element-wise addition of two matrices of the same dimensions. ✅

• Multiplication: Rows of the first matrix multiplied by columns of the second. ✅

• Inverse: A matrix that, when multiplied with the original, yields the identity matrix. ⚠️

Determinants

• 2x2 Determinant: For matrix A = [[a, b], [c, d]], det(A) = ad - bc. ✅

• 3x3 Determinant:

  det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
  

where A = [[a, b, c], [d, e, f], [g, h, i]]. ✅

Key Formulas

• Matrix Multiplication:

C = A B where C[i][j] = Σ A[i][k] B[k][j] for all k. ✅

• Inverse of 2x2 Matrix:

A^(-1) = (1/det(A)) * [[d, -b], [-c, a]] if det(A) ≠ 0. ⚠️

Common Mistakes to Avoid

Common Mistake: Confusing matrix multiplication with element-wise multiplication. Remember, they are different operations.

Memory Tricks

Tip: "A-M-I" — Addition, Multiplication, Inverse for matrix operations.

Key Diagrams

Matrix Example:
| a  b |
| c  d |
Determinant of 2x2:
det(A) = ad - bc
3x3 Determinant:
| a  b  c |
| d  e  f |
| g  h  i |