Gujarati: Basic I Back Matter I Appendix B: Rudiments of I I © The McGraw-Hill
Econometrics, Fourth Matrix Algebra Companies, 2004
APPENDIX B: RUDIMENTS OF MATRIX ALGEBRA 919
which is a matrix of order n x n. Note that the preceding matrix is symmetrical.
6. A matrix postmultiplied by a column vector is a column vector.
7. A row vector postmultiplied by a matrix is a row vector.
8. Matrix multiplication is associative; that is, (AB)C = A(BC), where A is M x N, B is N x P, and C is P x K.
9. Matrix multiplication is distributive with respect to addition; that is, A(B + C) = AB + AC and (B + C)A = BA + CA.
We have already defined the process of matrix transposition as interchanging the rows and the columns of a matrix (or a vector). We now state some of the properties of transposition.
1. The transpose of a transposed matrix is the original matrix itself. Thus, (A')' = A.
2. If A and B are conformable for addition, then C = A + B and C' = (A + B)' = A' + B'. That is, the transpose of the sum of two matrices is the sum of their transposes.
3. If AB is defined, then (AB)' = B'A'. That is, the transpose of the product of two matrices is the product of their transposes in the reverse order. This can be generalized: (ABCD)' = D'C'B'A'.
4. The transpose of an identity matrix I is the identity matrix itself; that is I' = I.
5. The transpose of a scalar is the scalar itself. Thus, if X is a scalar, X' = X.
6. The transpose of (XA)' is XA' where X is a scalar. [Note: (XA)' = A'X' = A'X = XA'.]
7. If A is a square matrix such that A = A', then A is a symmetric matrix. (Cf. the definition of symmetric matrix given previously.)
An inverse of a square matrix A, denoted by A 1 (read A inverse), if it exists, is a unique square matrix such that
where I is an identity matrix whose order is the same as that of A. For example
Was this article helpful?
Learning About The Rules Of The Rich And Wealthy Can Have Amazing Benefits For Your Life And Success. Discover the hidden rules and beat the rich at their own game. The general population has a love / hate kinship with riches. They resent those who have it, but spend their total lives attempting to get it for themselves. The reason an immense majority of individuals never accumulate a substantial savings is because they don't comprehend the nature of money or how it works.