Perform matrix operations including addition, multiplication, transpose, inverse, and determinant
Matrices must have the same dimensions. Operation is performed element-wise.
Example: [1,2; 3,4] + [5,6; 7,8] = [6,8; 10,12]
For A × B, the number of columns in A must equal the number of rows in B.
The resulting matrix has dimensions: (rows of A) × (columns of B)
Important: Matrix multiplication is NOT commutative (A×B ≠ B×A)
Flips a matrix over its diagonal. Rows become columns and vice versa.
If A is m×n, then A^T is n×m.
Only defined for square matrices. Indicates if matrix is invertible.
If det(A) = 0, the matrix is singular (not invertible).
For 2×2: det([a,b; c,d]) = ad - bc
The inverse A^(-1) exists only for square matrices with non-zero determinant.
Property: A × A^(-1) = I (identity matrix)
Used to solve linear systems: Ax = b → x = A^(-1)b
The rank is the dimension of the vector space spanned by the matrix columns (or rows).
It equals the number of linearly independent rows or columns.
Full rank: rank = min(rows, cols)