MMULT

Returns the ordinary product of two matrices.

Syntax:

MMULT(array1, array2)

multiplies array1 and array2, and returns the matrix result. array1 and array2 may each be either an inline array or a range, containing all numbers.

The number of columns in array1 must be the same as the number of rows in array2.

MMULT returns an array with the same number of rows as array1 and the same number of columns as array2.

Example:

MMULT({1,2,3},{4,5;6,7;8,9})

when entered as an array formula in cell B3, returns 40 in cell B3 and 46 in cell C3. A 1 x 3 matrix was multiplied by a 3 x 2 matrix, giving a 1 x 2 matrix result, calculated as 1*4 + 2*6 + 3*8 = 40 and 1*5 + 2*7 + 3*9 = 46.

Application:

Resource Allocation for Manufacturing

A company produces three types of products: Product A, Product B, and Product C. Each product requires a specific amount of raw materials (plastic, metal, and wood) and labor hours. The company has two different factories, Factory 1 and Factory 2, which produce these products.

Table 1: Resources Required per Product (Matrix R)

This table shows the resources required to produce a single unit of each product.

		Product A	Product B	Product C
	A	B	C	D
1	Plastic (kg)	2	3	4
2	Metal (kg)	5	1	2
3	Wood (kg)	1	4	3
4	Labor (hours)	3	2	5

Table 2: Production Volume by Factory (Matrix P)

This table shows the number of units of each product produced by each factory in a month.

		Factory 1	Factory 2
	A	B	C
1	Product A (units)	100	150
2	Product B (units)	80	120
3	Product C (units)	50	90

How MMULT is Used in this Example

The goal is to calculate the total amount of each resource (plastic, metal, wood, and labor) used by each factory. To do this, we multiply the Resources per Product matrix (Table 1) by the Production Volume matrix (Table 2).

The formula for matrix multiplication is: (Result Matrix) = (Resources Matrix) × (Production Matrix)

The dimensions must match for multiplication. The Resources Matrix (R) is a 4x3 matrix, and the Production Matrix (P) is a 3x2 matrix. The resulting Total Resources matrix will be a 4x2 matrix.

To find the total plastic used by Factory 1, we perform the following calculation: (2 kg/unit * 100 units) + (3 kg/unit * 80 units) + (4 kg/unit * 50 units) = 200 + 240 + 200 = 640 kg of plastic

This corresponds to multiplying the first row of Table 1 by the first column of Table 2.

We perform this type of calculation for each resource and each factory to fill the result matrix. The MMULT operation automates this process, providing the final result:

Table 3: Total Resources Used by Factory (Result Matrix)

		Factory 1	Factory 2
	A	B	C
1	Plastic (kg)	640	1100
2	Metal (kg)	670	1020
3	Wood (kg)	610	1050
4	Labor (hours)	710	1230

The MMULT function, in a general sense, performs this series of dot product calculations, taking two matrices as input and producing a single result matrix that represents the combined effect of the two initial matrices. In this example, it efficiently calculates the total resource consumption for each factory based on the resource requirements per product and the production volume of each product.