Matrix Multiplication Calculator

Matrix Multiplication is evaluated from A[1,1], A[1,2] and A[2,1]. The calculation reports AB[1,1], AB[1,2] and AB[2,1].

Results

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About the Matrix Multiplication Calculator

Matrix Multiplication is treated here as a quantitative relation between A[1,1], A[1,2], A[2,1] and A[2,2] and AB[1,1], AB[1,2], AB[2,1] and AB[2,2].

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
Each output element (i,j) is the sum of products of row i from A with column j from B. Requires matching inner dimensions. The commutative check verifies if AB = BA (rare for general matrices).

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

Each output element (i,j) is the sum of products of row i from A with column j from B. Requires matching inner dimensions. The commutative check verifies if AB = BA (rare for general matrices).

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Basic 2×2: [[1,2],[3,4]] × [[5,6],[7,8]]

Inputs

a11: 1 a12: 2 a21: 3 a22: 4 b11: 5 b12: 6 b21: 7 b22: 8
AB[1,1]: 19. AB[1,2]: 22. AB[2,1]: 43. AB[2,2]: 50. det: 4. AB = BA?: No - AB != BA (non-commutative)

With A[1,1] = 1, A[1,2] = 2, A[2,1] = 3 and A[2,2] = 4 as the stated inputs, the result is AB[1,1] = 19, AB[1,2] = 22 and AB[2,1] = 43. Each value corresponds to the declared output fields.

Example 2: Rotation compositions: R(45°) × R(45°) = R(90°)

Inputs

a11: 0.7071 a12: -0.7071 a21: 0.7071 a22: 0.7071 b11: 0.7071 b12: -0.7071 b21: 0.7071 b22: 0.7071
AB[1,1]: 0. AB[1,2]: -0.999981. AB[2,1]: 0.999981. AB[2,2]: 0. det: 0.999962. AB = BA?: Yes - AB = BA (commute)

With A[1,1] = 0.7071, A[1,2] = -0.7071, A[2,1] = 0.7071 and A[2,2] = 0.7071 as the stated inputs, the result is AB[1,1] = 0, AB[1,2] = -0.999981 and AB[2,1] = 0.999981. Each value corresponds to the declared output fields.

Example 3: Identity property: [[1,0],[0,1]] × [[3,5],[2,4]]

Inputs

a11: 1 a12: 0 a21: 0 a22: 1 b11: 3 b12: 5 b21: 2 b22: 4
AB[1,1]: 3. AB[1,2]: 5. AB[2,1]: 2. AB[2,2]: 4. det: 2. AB = BA?: Yes - AB = BA (commute)

With A[1,1] = 1, A[1,2] = 0, A[2,1] = 0 and A[2,2] = 1 as the stated inputs, the result is AB[1,1] = 3, AB[1,2] = 5 and AB[2,1] = 2. Each value corresponds to the declared output fields.

Example 4: Non-commutativity: [[1,2],[0,1]] × [[1,0],[3,1]] vs reversed

Inputs

a11: 1 a12: 2 a21: 0 a22: 1 b11: 1 b12: 0 b21: 3 b22: 1
AB[1,1]: 7. AB[1,2]: 2. AB[2,1]: 3. AB[2,2]: 1. det: 1. AB = BA?: No - AB != BA (non-commutative)

With A[1,1] = 1, A[1,2] = 2, A[2,1] = 0 and A[2,2] = 1 as the stated inputs, the result is AB[1,1] = 7, AB[1,2] = 2 and AB[2,1] = 3. Each value corresponds to the declared output fields.

Common Use Cases

  • Multiply two 2×2 matrices for linear algebra problems
  • Compute composed transformations in computer graphics
  • Solve linear algebra homework with shown steps