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Bnj Amn 52 which is a m × 1 matrix or column vector which equals A11 A1n A12 A21 A2n A22 .. B1j + .. B2j + · · · + .. . . . com Bnj . Linear Algebra I Matrices and Row operations Linear Transformations The ith entry of this m × 1 matrix is Ai1 B1j + Ai2 B2j + · · · + Ain Bnj = m ∑ Aik Bkj . 3. This motivates the deﬁnition for matrix multiplication which identiﬁes the ij th entries of the product. 8 Let A = (Aij ) be an m × n matrix and let B = (Bij ) be an n × p matrix.

When the multiplication is done it equals 13 13 29 32 . 0 0 Check this and be sure you come up with the same answer. 12 Multiply if possible 2 1 2 1 0 . 1 In this case you are trying to do (3 × 1) (1 × 4) . do it. 2 The inside numbers match so you can 1 2 2 4 1 2 1 2 1 0 0 0 Digraphs Consider the following graph illustrated in the picture. 2 1 3 There are three locations in this graph, labelled 1,2, and 3. The directed lines represent a way of going from one location to another.

Hence pi = qi for all i because it was already argued that p1 = q1 , and this results in a contradiction. 10 Systems Of Equations Sometimes it is necessary to solve systems of equations. For example the problem could be to ﬁnd x and y such that x + y = 7 and 2x − y = 8. 2) The set of ordered pairs, (x, y) which solve both equations is called the solution set. For example, you can see that (5, 2) = (x, y) is a solution to the above system. To solve this, note that the solution set does not change if any equation is replaced by a non zero multiple of itself.