2 Algebraic Manipulation of Matrices 2.1 Equality of matrices Matrices A and B are equal if and only if they have the same dimensions and each elements of A equal the corresponding element of B A=B if and only if aik= bik for all i and k 2.2 Transposition The transpose of a matrix A, denoted A, is obtained by creating the matrix whose kth row is the kth column of the original matrix. If A is i x k, a is k x For example 1563 A 645 A 2141 314 3554 If A is symmetric, A=A. It is also apparent that for any matrix A, (A)=A Finally,the transpose of a column vector, a, is a row vector: 2.3 Matrix Addition Matrices cannot be added unless they have the same dimension. The operation of addition is extended to matrices by definin A+B=aik + bil We also extend the operation of subtraction to matrices precisely as if they were scalars by performing the operation element by element. Thus It follows that matrix addition is commutative A+b=b+A and associative2 Algebraic Manipulation of Matrices 2.1 Equality of Matrices Matrices A and B are equal if and only if they have the same dimensions and each elements of A equal the corresponding element of B. A=B if and only if aik = bik for all i and k. 2.2 Transposition The transpose of a matrix A, denoted A′ , is obtained by creating the matrix whose kth row is the kth column of the original matrix. If A is i × k, A′ is k × i. For example, A =     1 2 3 5 1 5 6 4 5 3 1 4     , A′ =   1 5 6 3 2 1 4 1 3 5 5 4  . If A is symmetric, A=A′ . It is also apparent that for any matrix A, (A′ ) ′ = A. Finally, the transpose of a column vector, ai is a row vector: a ′ i = a1 a2 . . . ai . 2.3 Matrix Addition Matrices cannot be added unless they have the same dimension. The operation of addition is extended to matrices by defining C = A + B=[aik + bik]. We also extend the operation of subtraction to matrices precisely as if they were scalars by performing the operation element by element. Thus, C = A − B=[aik − bik]. It follows that matrix addition is commutative, A + B = B + A, and associative, 2