Concept Architecture
Concept
Theoretically, Matrix Multiplication is the fundamental operation of matrix algebra in which two compatible matrices are combined to produce a new matrix representing the composition of two linear transformations. Unlike ordinary multiplication, matrix multiplication is generally non-commutative, meaning that the order of multiplication affects the result. The concept is foundational to linear algebra, numerical analysis, econometrics and scientific computing, and underlies nearly every computational method used in health economics.
Mathematically, Matrix Multiplication is performed by calculating the dot product of each row of the first matrix with each column of the second matrix. The operation is defined only when the number of columns of the first matrix equals the number of rows of the second matrix. The resulting matrix contains elements computed as the sum of products of corresponding row and column entries.
In practice, Matrix Multiplication is used extensively throughout health economics in regression estimation, Markov state-transition models, covariance calculations, optimisation algorithms, machine learning, probabilistic sensitivity analysis and multivariate statistical methods. Efficient matrix multiplication is fundamental to virtually all numerical software used in health economic modelling.
Purpose
Used to combine linear transformations, solve systems of equations, perform multivariate statistical analyses and support computational methods used throughout health economics.
Mathematical Formulae
Primary Formula
If
A = (a??)
and
B = (b??)
then
C = AB
where
c?? = �? a??b??
Supporting Formulae
Dimension Requirement
(m ? n)(n ? p) = (m ? p)
Identity Property
AI = IA = A
Related Mathematical Methods
- Matrix algebra
- Matrix inversion
- LU decomposition
- QR decomposition
- Singular value decomposition
- Eigenvalue decomposition
Example
Consider
A =
[ \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} ]
and
B =
[ \begin{bmatrix} 2 & 1 \ 0 & 5 \end{bmatrix} ]
Then
AB =
[ \begin{bmatrix} (1?2)+(2?0) & (1?1)+(2?5) \ (3?2)+(4?0) & (3?1)+(4?5) \end{bmatrix}
\begin{bmatrix} 2 & 11 \ 6 & 23 \end{bmatrix} ]
The resulting matrix represents the combined effect of the two linear transformations.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(A2:B3,D2:E3) | Perform matrix multiplication for regression, Markov and optimisation models. |
| TRANSPOSE | =TRANSPOSE(A2:B3) | Prepare matrices for compatible multiplication. |
| SUMPRODUCT | =SUMPRODUCT(A2:B2,C2:C3) | Calculate individual row-column products. |
| MUNIT | =MUNIT(2) | Generate identity matrices for verification. |
| LET/LAMBDA | Custom matrix algorithms | Implement advanced matrix multiplication procedures in Excel. |
VBA (Optional)
Automate matrix multiplication for regression estimation, state-transition modelling, covariance analysis and numerical optimisation within health economic models.
Sources
- Strang G. Linear Algebra and Its Applications.
- Golub GH, Van Loan CF. Matrix Computations.
- Trefethen LN, Bau D. Numerical Linear Algebra.
- Higham NJ. Accuracy and Stability of Numerical Algorithms.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
Related Concepts (6)
Library
Publications
1
Accuracy and Stability of Numerical Algorithms — Nicholas J. Higham, 2nd Edition ed., 2002 (Society for Industrial and Applied Mathematics (SIAM))
The standard reference on floating-point error, conditioning, stability and the numerical reliability of matrix, polynomial and linear-algebra algorithms.
BookView source →
Frequently Asked Questions (6)
What is matrix multiplication?
An operation that combines two compatible matrices by calculating the dot products of their rows and columns.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
How does matrix multiplication combine two matrices?
Matrix multiplication combines two compatible matrices by calculating the dot products of the rows of the first with the columns of the second. Each entry of the result comes from one such dot product, pairing a row with a column. This construction of the product from row-by-column dot products is how matrix multiplication combines the two matrices Because each entry is a row-by-column dot product, the two matrices must have matching inner dimensions, and the operation underlies much of how transformations and systems are composed.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
Why must matrices be compatible for matrix multiplication?
Matrices must be compatible for matrix multiplication because the operation pairs each row of the first matrix with each column of the second through a dot product, which requires the row and column to have matching lengths. If the dimensions do not align, the dot products cannot be formed. This need for compatible dimensions follows directly from how matrix multiplication is defined Because each entry is a row-by-column dot product, the two matrices must have matching inner dimensions, and the operation underlies much of how transformations and systems are composed.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What does each entry of the result come from in matrix multiplication?
In matrix multiplication, each entry of the result comes from the dot product of a row of the first matrix with a column of the second. The entry's position reflects which row and which column were combined. This origin of every entry in a single row-by-column dot product is basic to how matrix multiplication produces its result Because each entry is a row-by-column dot product, the two matrices must have matching inner dimensions, and the operation underlies much of how transformations and systems are composed.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What are the inputs to matrix multiplication?
The inputs to matrix multiplication are two compatible matrices, combined by calculating the dot products of the rows of one with the columns of the other. Their compatibility, meaning aligned dimensions, is required for the operation to proceed. Taking two such matrices as inputs is part of what defines matrix multiplication Because each entry is a row-by-column dot product, the two matrices must have matching inner dimensions, and the operation underlies much of how transformations and systems are composed.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
How does matrix multiplication relate to matrix algebra?
Matrix multiplication is an operation that combines two compatible matrices through the dot products of their rows and columns, while matrix algebra is the branch of mathematics concerned with representing and manipulating matrices and the operations performed on them. Matrix multiplication is one of the operations that matrix algebra defines and studies. The two are connected as a specific operation and the field that encompasses it Because each entry is a row-by-column dot product, the two matrices must have matching inner dimensions, and the operation underlies much of how transformations and systems are composed.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 26 May 2026
Content version: 1.0.0
Canonical Identity
- Term code
- CS-LA-MO-002
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