What does determinant = 0 mean?

The matrix is singular (not invertible). The transformation collapses space to a lower dimension. The rows/columns are linearly dependent.

Can determinant be negative?

Yes. Negative determinant means the transformation includes a reflection (flips orientation). The absolute value still gives the scaling factor.

Use LU decomposition: det(A) = det(L)×det(U) = product of diagonal entries. O(n³) vs O(n!) for cofactor expansion.

Yes, transposing twice returns the original matrix. Transpose is its own inverse operation.

A matrix where A = Aᵀ. Covariance matrices, distance matrices, and adjacency matrices of undirected graphs are symmetric.

det(Aᵀ) = det(A). The determinant is unchanged by transpose.

No, matrices must have identical dimensions for addition. Broadcasting (like NumPy) is a separate operation that expands dimensions.

Yes: (A + B) + C = A + (B + C). You can group additions in any order.

A matrix of all zeros. A + 0 = A for any matrix A of the same dimensions.

Yes: A - B = A + (-B). This is useful for understanding properties.

The zero matrix. Any matrix minus itself gives all zeros.

Yes, individual elements can be negative even if inputs are positive.

Multiply, add, subtract, transpose matrices and calculate determinants with step-by-step solutions.

Operation Being Performed

C=A × B(2×3) × (3×2) = 2×2

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Quick Reference:

A×B ≠ B×A in general. The dimensions must be compatible (cols of A = rows of B), and even when both orders work, the results usually differ.

Naive multiplication is O(n³). Strassen's algorithm is O(n^2.807). Modern GPUs parallelize this efficiently for deep learning.

Only when cols(A) = rows(B). An m×n matrix times an n×p matrix gives an m×p result.