How to Do Matrix Operations — Multiply, Inverse, Determinant & Rank

Core Matrix Operations and Their Rules

A matrix is a rectangular array of numbers arranged in rows and columns. The size is denoted as m × n (m rows, n columns). Operations depend on dimensions and properties.

1. Matrix Multiplication

Matrix multiplication is not commutative (AB ≠ BA) and requires the inner dimensions to match.

Rule:
For A (m × n) and B (n × p), the product C = AB is (m × p).

Formula:
C[i][j] = Σ (A[i][k] × B[k][j]) for k = 1 to n

Example:
A = [[1, 2], [3, 4]] (2×2)
B = [[5, 6], [7, 8]] (2×2)
C = AB = [[1×5 + 2×7, 1×6 + 2×8], [3×5 + 4×7, 3×6 + 4×8]] = [[19, 22], [43, 50]]

Key Insight:

• AB is defined only if A’s columns = B’s rows
• Resulting matrix has A’s rows and B’s columns

2. Determinant (for Square Matrices)

The determinant (det(A) or |A|) is a scalar value that indicates whether a matrix is invertible (det ≠ 0) and describes scaling factors in linear transformations.

2×2 Formula:
det([[a, b], [c, d]]) = ad – bc

3×3 Formula (Rule of Sarrus or Cofactor Expansion):
det = a(ei − fh) − b(di − fg) + c(dh − eg) for matrix [[a,b,c],[d,e,f],[g,h,i]]

Example:
A = [[2, 3], [1, 4]] → det = 2×4 – 3×1 = 8 – 3 = 5 (invertible)

3. Matrix Inverse (A⁻¹)

The inverse exists only for square, non-singular matrices (det ≠ 0), and satisfies AA⁻¹ = I (identity matrix).

2×2 Inverse Formula:
A⁻¹ = (1/det(A)) × [[d, -b], [-c, a]] for A = [[a, b], [c, d]]

Example:
A = [[2, 3], [1, 4]], det = 5
A⁻¹ = (1/5) × [[4, -3], [-1, 2]] = [[0.8, -0.6], [-0.2, 0.4]]

For larger matrices, use Gauss-Jordan elimination or adjugate method.

4. Solving Ax = b (Linear Systems)

This represents a system of linear equations. Solutions exist if b is in the column space of A.

Methods:

• Inverse method: x = A⁻¹b (if A is invertible)
• Gaussian elimination: Row-reduce [A | b] to [I | x]
• Cramer’s Rule: For small systems, xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i with b

Example:
2x + 3y = 8
x + 4y = 7
Matrix form: [[2,3],[1,4]] [x,y]ᵀ = [8,7]ᵀ
Solution: x = 1, y = 2

5. Other Key Operations

• Transpose (Aᵀ): Flip rows and columns
• Rank: Number of linearly independent rows/columns (max rank = min(m,n))
• Trace: Sum of diagonal elements (for square matrices)
• Eigenvalues/Eigenvectors: Solve Av = λv (advanced topic)

Step-by-Step: Gauss-Jordan Elimination for Ax = b

This robust method works for any system (consistent or inconsistent).

1. Form augmented matrix [A | b]
2. Forward elimination: Use row operations to create zeros below pivots
3. Back substitution: Create zeros above pivots to get reduced row echelon form (RREF)
4. Read solution: If RREF is [I | x], then x is the solution

Example:
Solve x + y = 5, 2x – y = 1
Augmented matrix: [[1, 1, 5], [2, -1, 1]]
→ RREF: [[1, 0, 2], [0, 1, 3]] → x = 2, y = 3

Pro Tips & Common Mistakes

• Check dimensions first: Multiplication fails if inner dimensions don’t match
• Determinant ≠ 0 for inverse: Always compute det before attempting inversion
• Use row operations carefully: Scale rows, swap rows, add multiples—never multiply rows
• Verify solutions: Plug x back into Ax to confirm it equals b
• Software for large matrices: For 4×4+, use tools like NumPy, MATLAB, or this calculator

Practical Applications

• Computer Graphics: Transformation matrices (rotation, scaling, translation)
• Machine Learning: Data represented as matrices; operations in neural networks
• Engineering: Solving circuit equations (Kirchhoff’s laws) or structural loads
• Economics: Input-output models (Leontief models)
• Cryptography: Hill cipher uses matrix multiplication for encryption