Concept Architecture
Concept
Theoretically, Matrix Inversion is the process of determining the inverse of a non-singular square matrix such that multiplication of the matrix by its inverse yields the identity matrix. It is a fundamental operation in linear algebra and numerical analysis, providing the mathematical basis for solving systems of linear equations, estimating regression coefficients and performing matrix transformations. Matrix inversion is extensively used throughout health economics in statistical estimation, optimisation and decision modelling.
Mathematically, the inverse exists only for square matrices with non-zero determinants. The inverse matrix satisfies the defining property that multiplication in either order produces the identity matrix. Numerical algorithms such as Gaussian elimination, LU decomposition and Cholesky decomposition are commonly used to compute matrix inverses efficiently while maintaining numerical stability for large systems.
In practice, Matrix Inversion is widely applied in health economics for ordinary least squares regression, generalised linear models, covariance matrix analysis, Markov modelling, Bayesian estimation and calibration of decision models. Modern statistical software performs matrix inversion automatically during estimation procedures, making it a foundational computational operation in applied health economic research.
Purpose
Used to solve systems of linear equations, estimate statistical models, perform matrix transformations and support numerical optimisation and computational methods within health economics.
Mathematical Formulae
Primary Formula
AA?? = A??A = I
where:
- A = non-singular square matrix
- A?? = inverse matrix
- I = identity matrix
Supporting Formulae
For a 2 ? 2 matrix
A =
[ \begin{bmatrix} a & b \ c & d \end{bmatrix} ]
A?? = (1 � (ad ? bc))
[ \begin{bmatrix} d & -b \ -c & a \end{bmatrix} ]
provided
det(A) ? 0.
Related Mathematical Methods
- Gaussian elimination
- LU decomposition
- Cholesky decomposition
- QR decomposition
- Matrix algebra
- Determinant analysis
Example
Consider the matrix
A =
[ \begin{bmatrix} 4 & 7 \ 2 & 6 \end{bmatrix} ]
The determinant is
det(A) = (4 ? 6) ? (7 ? 2)
= 10
The inverse is
A?? =
[ \frac{1}{10} \begin{bmatrix} 6 & -7 \ -2 & 4 \end{bmatrix} ]
Multiplying A by A?? produces the identity matrix, confirming that the inverse has been correctly calculated.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MINVERSE | =MINVERSE(A2:B3) | Calculate the inverse of a square matrix. |
| MMULT | =MMULT(A2:B3,C2:D3) | Verify that A ? A?? equals the identity matrix. |
| MDETERM | =MDETERM(A2:B3) | Confirm that the determinant is non-zero before inversion. |
| MUNIT | =MUNIT(2) | Generate the identity matrix for verification. |
| LET/LAMBDA | Custom numerical algorithms | Implement advanced matrix inversion procedures in Excel. |
VBA (Optional)
Automate matrix inversion for regression estimation, covariance analysis, optimisation routines and decision-analytic 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 (5)
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 inversion?
The process of finding a matrix that, when multiplied by the original matrix, produces the identity matrix.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What does matrix inversion produce?
Matrix inversion produces a matrix that, when multiplied by the original matrix, produces the identity matrix. This resulting matrix is the inverse, and multiplying it by the original in either order returns the identity. Finding this inverse, the matrix that undoes the effect of the original under multiplication, is what matrix inversion delivers Because the inverse undoes the original matrix under multiplication, finding it is a way to solve linear systems, though only matrices that admit such a partner can be inverted at all.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What condition does the result of matrix inversion satisfy?
The result of matrix inversion satisfies the condition that, multiplied by the original matrix, it produces the identity matrix. This defining relation is what marks the found matrix as the inverse of the original. Meeting this condition of yielding the identity when multiplied by the original is what makes matrix inversion successful Because the inverse undoes the original matrix under multiplication, finding it is a way to solve linear systems, though only matrices that admit such a partner can be inverted at all.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
Why is the identity matrix central to matrix inversion?
The identity matrix is central to matrix inversion because the process seeks the matrix that, multiplied by the original, produces exactly the identity. The identity plays the role of a neutral element under multiplication, so reaching it confirms the inverse has been found. This target of producing the identity matrix defines the goal of matrix inversion Because the inverse undoes the original matrix under multiplication, finding it is a way to solve linear systems, though only matrices that admit such a partner can be inverted at all.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
Can every matrix undergo matrix inversion?
Not every matrix can undergo matrix inversion, since only a matrix for which another matrix exists that multiplies with it to produce the identity has an inverse. Matrices lacking such a partner cannot be inverted. This restriction, following from the requirement to yield the identity matrix, means matrix inversion applies only to invertible matrices Because the inverse undoes the original matrix under multiplication, finding it is a way to solve linear systems, though only matrices that admit such a partner can be inverted at all.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
How does matrix inversion relate to matrix multiplication?
Matrix inversion is the process of finding a matrix that, multiplied by the original, produces the identity, while matrix multiplication is the operation that combines two compatible matrices through the dot products of their rows and columns. Matrix inversion is defined through matrix multiplication, since the inverse is identified by the product it forms. The two are connected, as inversion seeks the matrix giving the identity under multiplication Because the inverse undoes the original matrix under multiplication, finding it is a way to solve linear systems, though only matrices that admit such a partner can be inverted at all.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 26 May 2026
Content version: 1.0.0
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- Persistent URI
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- Term code
- CS-LA-MO-003
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