How Matrix Multiplication Works
Matrix multiplication combines two matrices to produce a third. Unlike regular multiplication, order matters. A × B usually gives a different result than B × A, and sometimes one is possible while the other isn’t.
Each element in the result comes from a dot product. For position [i,j] in the result, take row i from matrix A and column j from matrix B, multiply corresponding elements, then sum them.
Operation Reference
Combines transformations, solves systems of equations, applies weights in neural networks. Result dimensions: A’s rows × B’s columns.
Element-by-element operation. Used for combining datasets, calculating differences between states, superimposing effects.
Rows become columns, columns become rows. Element A[i,j] moves to position [j,i]. Essential for many formulas.
Single number revealing key properties. Zero means singular (no inverse). Non-zero means the transformation is reversible.
Matrix equivalent of division. A × A⁻¹ = Identity. Used to solve AX = B by computing X = A⁻¹B.
Transpose Visualization
The transpose operation flips a matrix over its diagonal. Every row becomes a column, every column becomes a row.
The Identity Matrix
The identity matrix has 1s on the diagonal and 0s everywhere else. It functions like the number 1 in regular multiplication: any matrix multiplied by the identity returns unchanged.
Determinant: 2×2 Formula
For a 2×2 matrix, the determinant follows a simple cross-multiplication pattern:
Singular Matrices
A matrix with determinant zero is called singular. It represents a transformation that collapses at least one dimension, losing information that cannot be recovered. Singular matrices have no inverse.
The determinant equals zero, so no inverse exists. This happens when rows or columns are linearly dependent (one row/column is a multiple of another, or can be created by combining others).
Practical Applications
Every rotation, scaling, and translation in 3D graphics is a matrix multiplication. Games chain dozens of transformations per frame.
Neural networks are chains of matrix multiplications with activation functions. Training involves transposes and gradients.
Any system like 2x + 3y = 7 can be written as AX = B. Solving means computing X = A⁻¹B.
Structural analysis models forces across elements. Circuit analysis and control systems use matrix equations extensively.
Common Mistakes
Matrix Properties Reference
Verifying Your Matrix Calculation
For multiplication: Check that result dimensions are correct. Verify a few elements manually using the dot product method shown above.
For inverse: Multiply your result by the original matrix. You should get the identity matrix (or very close, accounting for rounding).
For determinant: For 2×2, use ad − bc. For larger matrices, expansion along any row or column should give the same value.
When multiplying matrices of all positive numbers, the result should also contain positive numbers. If you see unexpected negatives, check your arithmetic or input values.