How do I compute a single element of the product matrix?

Element (i,j) of A×B is the dot product of row i of A with column j of B: sum of A[i][k]×B[k][j] for all k.

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Q: When can two matrices be multiplied?

Matrix A (m×n) can multiply matrix B (n×p) only when the number of columns in A equals the number of rows in B. The result has dimensions m×p.

Q: Is matrix multiplication commutative?

No. In general A×B does not equal B×A. Matrix multiplication is associative (A(BC)=(AB)C) and distributive (A(B+C)=AB+AC), but NOT commutative.

Q: What is the identity matrix for multiplication?

The identity matrix I (1s on diagonal, 0s elsewhere) satisfies A×I = I×A = A. It acts like the number 1 for matrix multiplication.

How Matrix Multiplication Works

Each element (i,j) in the product matrix C = A×B is computed as the dot product of row i of A with column j of B:

C[i][j] = A[i][0]×B[0][j] + A[i][1]×B[1][j] + ... + A[i][n-1]×B[n-1][j]

For this to work, the number of columns in A must equal the number of rows in B. If A is m×n and B is n×p, then C = A×B is m×p.

Worked Example

Multiply A (2×2) by B (2×2):

A = [1, 2] B = [5, 6]
[3, 4] [7, 8]

C[0][0] = 1×5 + 2×7 = 19
C[0][1] = 1×6 + 2×8 = 22
C[1][0] = 3×5 + 4×7 = 43
C[1][1] = 3×6 + 4×8 = 50

A×B = [19, 22] [43, 50]

Key Properties

NOT commutative: A×B ≠ B×A in general
Associative: A×(B×C) = (A×B)×C
Distributive: A×(B+C) = A×B + A×C
Identity: A×I = I×A = A (where I is the identity matrix)
Zero matrix: A×O = O×A = O

Dimension Rule

Only compatible dimensions can be multiplied:

2×3 matrix × 3×4 matrix = 2×4 matrix (valid)
2×3 matrix × 2×4 matrix = undefined (invalid: 3 ≠ 2)
1×3 row vector × 3×1 column vector = 1×1 scalar (dot product)

Real-World Applications

Computer graphics: Rotation, scaling, and translation transformations are composed by multiplying transformation matrices.
Machine learning: Neural network layers compute y = W×x + b where W is the weight matrix.
Economics: Input-output models use matrix multiplication to track how industries depend on each other.
Cryptography: Hill cipher encryption multiplies plaintext vectors by a key matrix.

Frequently Asked Questions

When can two matrices be multiplied?

Matrix A (m×n) can multiply B (n×p) only when cols(A) = rows(B). The result is m×p.

Is matrix multiplication commutative?

No. A×B ≠ B×A in general. It is associative and distributive, but not commutative.

What is the identity matrix for multiplication?

The identity matrix I has 1s on the diagonal and 0s elsewhere. A×I = I×A = A.

How do I compute a single element of the product?

Element C[i][j] = dot product of row i of A with column j of B = sum of A[i][k]×B[k][j] for all k.