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Gujarati: Basic I Back Matter I Appendix B: Rudiments of I I © The McGraw-Hill

Econometrics, Fourth Matrix Algebra Companies, 2004

Edition

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.

Matrix Transposition

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.)

Matrix Inversion

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

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