Pairwise Matrix Product Computation

Medium

Matrix multiplication is one of the most fundamental operations in linear algebra and serves as the backbone for countless algorithms in machine learning, computer graphics, physics simulations, and data science. When two matrices are multiplied together, the result encodes a composition of linear transformations, enabling complex multi-step transformations to be expressed as a single matrix operation. Given two matrices A of dimensions n × m and B of dimensions p × q, the product C = A · B is defined only when the number of columns in A equals the number of rows in B (i.e., m = p). When this condition is satisfied, the resulting matrix C has dimensions n × q. Each element C[i][j] of the resulting matrix is computed as the dot product of the i-th row of matrix A and the j-th column of matrix B: $$C_{ij} = \sum_{k=1}^{m} A_{ik} \cdot B_{kj}$$ Dimensional Compatibility Rule: Matrix A: n × m (n rows, m columns) Matrix B: p × q (p rows, q columns) For multiplication to be valid: m must equal p Resulting matrix C: n × q (n rows, q columns) Important Properties: Matrix multiplication is associative: (A · B) · C = A · (B · C) Matrix multiplication is NOT commutative: A · B ≠ B · A in general The identity matrix I acts as a neutral element: A · I = I · A = A Your Task: Write a Python function that computes the product of two matrices. The function should: Return a 2D list representing the resulting matrix if the dimensions are compatible Return -1 if the matrices cannot be multiplied (i.e., the number of columns in the first matrix does not equal the number of rows in the second matrix)

Example 1

Input

a = [[1, 2], [2, 4]]
b = [[2, 1], [3, 4]]

Output

[[8, 9], [16, 18]]

Explanation

Matrix A is 2×2 with 2 columns. Matrix B is 2×2 with 2 rows. Since columns(A) = rows(B) = 2, multiplication is valid. The result is a 2×2 matrix computed as: • C[0][0] = (1×2) + (2×3) = 2 + 6 = 8 • C[0][1] = (1×1) + (2×4) = 1 + 8 = 9 • C[1][0] = (2×2) + (4×3) = 4 + 12 = 16 • C[1][1] = (2×1) + (4×4) = 2 + 16 = 18 The resulting product matrix is [[8, 9], [16, 18]].

Example 2

Input

a = [[1, 2], [2, 4]]
b = [[2, 1], [3, 4], [4, 5]]

Output

-1

Explanation

Matrix A is 2×2 with 2 columns. Matrix B is 3×2 with 3 rows. Since columns(A) = 2 but rows(B) = 3, and 2 ≠ 3, the dimensions are incompatible for matrix multiplication. The function returns -1.

Example 3

Input

a = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
b = [[5, 6, 7], [8, 9, 10], [11, 12, 13]]

Output

[[5, 6, 7], [8, 9, 10], [11, 12, 13]]

Explanation

Matrix A is the 3×3 identity matrix. The identity matrix has 1s on the main diagonal and 0s elsewhere. Multiplying any matrix by the identity matrix returns the original matrix unchanged. • I · B = B (identity is the neutral element for matrix multiplication) This demonstrates that the identity matrix preserves any matrix it multiplies, just as multiplying a number by 1 preserves that number.

Constraints

1 ≤ n, m ≤ 100 (dimensions of matrix A)
1 ≤ p, q ≤ 100 (dimensions of matrix B)
-10⁶ ≤ A[i][j] ≤ 10⁶
-10⁶ ≤ B[i][j] ≤ 10⁶
Both matrices will have at least one row and one column
All rows within each matrix will have the same length (valid matrix format)

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Pairwise Matrix Product Computation

Medium

Matrix multiplication is one of the most fundamental operations in linear algebra and serves as the backbone for countless algorithms in machine learning, computer graphics, physics simulations, and data science. When two matrices are multiplied together, the result encodes a composition of linear transformations, enabling complex multi-step transformations to be expressed as a single matrix operation. Given two matrices A of dimensions n × m and B of dimensions p × q, the product C = A · B is defined only when the number of columns in A equals the number of rows in B (i.e., m = p). When this condition is satisfied, the resulting matrix C has dimensions n × q. Each element C[i][j] of the resulting matrix is computed as the dot product of the i-th row of matrix A and the j-th column of matrix B: $$C_{ij} = \sum_{k=1}^{m} A_{ik} \cdot B_{kj}$$ Dimensional Compatibility Rule: Matrix A: n × m (n rows, m columns) Matrix B: p × q (p rows, q columns) For multiplication to be valid: m must equal p Resulting matrix C: n × q (n rows, q columns) Important Properties: Matrix multiplication is associative: (A · B) · C = A · (B · C) Matrix multiplication is NOT commutative: A · B ≠ B · A in general The identity matrix I acts as a neutral element: A · I = I · A = A Your Task: Write a Python function that computes the product of two matrices. The function should: Return a 2D list representing the resulting matrix if the dimensions are compatible Return -1 if the matrices cannot be multiplied (i.e., the number of columns in the first matrix does not equal the number of rows in the second matrix)

Example 1

Input

a = [[1, 2], [2, 4]]
b = [[2, 1], [3, 4]]

Output

[[8, 9], [16, 18]]

Explanation

Example 2

Input

a = [[1, 2], [2, 4]]
b = [[2, 1], [3, 4], [4, 5]]

Output

-1

Explanation

Matrix A is 2×2 with 2 columns. Matrix B is 3×2 with 3 rows. Since columns(A) = 2 but rows(B) = 3, and 2 ≠ 3, the dimensions are incompatible for matrix multiplication. The function returns -1.

Example 3

Input

a = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
b = [[5, 6, 7], [8, 9, 10], [11, 12, 13]]

Output

[[5, 6, 7], [8, 9, 10], [11, 12, 13]]

Explanation

Constraints

1 ≤ n, m ≤ 100 (dimensions of matrix A)
1 ≤ p, q ≤ 100 (dimensions of matrix B)
-10⁶ ≤ A[i][j] ≤ 10⁶
-10⁶ ≤ B[i][j] ≤ 10⁶
Both matrices will have at least one row and one column
All rows within each matrix will have the same length (valid matrix format)