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Matrix Multiplication
Inverse Matrix


A matrix is composed of several rows and columns of numbers:


As shown in [1], a m x n (dimension) matrix has m rows and n columns. The elements of the matrix are denoted by aij where i = the row number, and j = the column number. In biomechanics and motion analysis, 3 x 3 and 3 x 1 matrices are the most commonly used types.

A matrix of m = n is called a square matrix. In a diagonal matrix which is a square matrix, all elements except the diagonal ones (i = j) are zero:


Matrices of the same dimension can be added or subtracted, element by element:


As shown in [2], matrix addition is commutative.

When a matrix is multiplied by a scalar, all the elements of the matrix magnify by the scalar value:


This operation is commutative as shown in [4].

Transpose of a m x n matrix is a n x m matrix whose columns are identical to the corresponding rows of the original matrix:


Note in [5] that the superscripted t , ()t, is used as the symbol for the transpose. Transposing suffices the following:



Matrix Multiplication

One of the most useful properties of the matrix is the matrix multiplication. Imagine a system of linear equations:


where a's & c's = scalars, and x, y & z = the unknowns. [7] can be simplified as:


[8] is expressed in matrix multiplication form. The general form of matrix multiplication is




Note that the number of columns in the left vector (n in [9]) must be the same to the number of rows in the right matrix. The dimension of the resulting vector is m (rows of the left matrix) x p (columns of the right matrix) as shown in [9].

The matrix multiplication is not commutative:


 but is distributive:




Matrix multiplication is also associative:


For any scalar d:


An identity matrix (I) is a square matrix whose diagonal elements are all 1 while the off-diagonal elements are all 0:




where d shown in [17] is the Kronecker delta:


From [5] and [9]:




The determinant of a 2 x 2 matrix is defined as:


The determinant of a 3 x 3 matrix can be reduced to a series of the determinants of 2 x 2 matrices:


To generalize [21] for any square matrix, a new matrix needs to be defined:


In other words, matrix aij is matrix a less the i-th row and the j-th column. Then, [21] can be generalize to


[23] is called a cofactor expansion across the first row of a. In fact, the determinant of a can be cofactor-expanded across any row or column:


The following properties of the determinant hold:



Inverse matrix

The inverse matrix of a square matrix suffices the following relationship:


where a-1 = the inverse matrix of a, and I = the identity matrix. For a square matrix to be invertible, its determinant must not be 0. The inverse matrix of square matrix a can be expressed as


The matrix of cofactors of a on the right side in [27] is called the adjugate of a. Note in [27] that element aij of the adjugate is associated with det(aji) rather than det(aij).

A system of linear equations such as that in [8] can be generalized as


where a = the known square coefficient matrix, b = the known column matrix, and x = the unknown parameter matrix. If matrix a is invertible, the unknown parameters can be obtained as follows:


In motion analysis, matrix a shown in [28] is generally not square. If matrix a is not a square matrix, the system of linear equations can be solved as


[30] is the so-called least square method. See Least Square Method for more details.



Young-Hoo Kwon, 1998-