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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition D 5 Copyright © Orchard Publications Matrix Operations For matrix multiplication, the operation is row by column. Thus, to obtain the product , we multiply each element of a row of by the corresponding element of a column of ; then, we add these products. Example D.3 Matrices and are defined as and Compute the products and Solution: The dimensions of matrices and are respectively ; therefore the product is feasible, and will result in a , that is, The dimensions for and are respectively and therefore, the product is also feasible. Multiplication of these will produce a matrix as follows: Check with MATLAB: C=[2 3 4]; D=[1 1 2]’; % Define matrices C and D. Observe that D is a column vector C*D, D*C % Multiply C by D, then multiply D by C ans = 7 Here, B and A are not conformable for multiplication B A p × n m × p A B A B C D C 2 3 4 = D 1 1 2 = C D D C C D 1 3 3 1 × × C D 1 1 × C D 2 3 4 1 1 2 2 ( ) 1 ( ) 3 ( ) 1 ( ) 4 ( ) 2 ( ) + + 7 = = = D C 3 1 1 3 × × D C 3 3 × D C 1 1 2 2 3 4 1 ( ) 2 ( ) 1 ( ) 3 ( ) 1 ( ) 4 ( ) 1 ( ) 2 ( ) 1 ( ) 3 ( ) 1 ( ) 4 ( ) 2 ( ) 2 ( ) 2 ( ) 3 ( ) 2 ( ) 4 ( ) 2 3 4 2 3 4 4 6 8 = = =

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Appendix D Matrices and Determinants D 6 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications ans = 2 3 4 -2 -3 -4 4 6 8 Division of one matrix by another, is not defined. However, an analogous operation exists, and it will become apparent later in this chapter when we discuss the inverse of a matrix. D.3 Special Forms of Matrices A square matrix is said to be upper triangular when all the elements below the diagonal are zero. The matrix of (D.4) is an upper triangular matrix. In an upper triangular matrix, not all elements above the diagonal need to be non zero. (D.4) A square matrix is said to be lower triangular , when all the elements above the diagonal are zero. The matrix of (D.5) is a lower triangular matrix. In a lower triangular matrix, not all elements below the diagonal need to be non zero.
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