Math 1b — Matrix MultiplicationIf A has rows aiand B has columns bj,thenAB has, by definition, aibjas the entryin row i and column j.ThematrixAB is the matrix of dot products of rows of A withcolumns of B.Here are some simple properties and facts about matrix multiplication. These rulesfollow directly from the definition of matrix multiplication. Small examples can helpunderstanding.1. A (row) vector times a matrix is a linear combination of the rows of that matrix(and the coefficients are the entries of the vector):( c1c2... c)⎛⎜⎜⎝—a1——a2—...—a—⎞⎟⎟⎠= c1a1+ c2a2+ ...+ ca.2. The rows of the matrix product AB are (rows of A)timesB.⎛⎜⎜⎝— a1—— a2—...— a—⎞⎟⎟⎠B =⎛⎜⎜⎝— a1B —— a2B —...— aB —⎞⎟⎟⎠.[From (1) and (2), we can see that if the rows of a matrix C are linear combinations of therows of B,thenC = AB for some matrix A.InthecasethatC is row-equivalent to B,we will show that there is such a matrix A which is a product of ”elementary matrices”.]3. A matrix times a (column) vector is a linear combination of the columns of thatmatrix (and the coefficients are the entries of the vector).⎛⎝|| |b1b2... bk|| |⎞⎠⎛⎜⎜⎝c1c2...ck⎞⎟⎟⎠= c1⎛⎝|b1|⎞⎠+ c2⎛⎝|b2|⎞⎠+ ...+ ck⎛⎝|bk|⎞⎠.4. The columns of AB are A times (columns of B).A⎛⎝|| |b1b2... bk|| |⎞⎠=⎛⎝|| |Ab1Ab2... Abk|| |⎞⎠.It is clear from this thatA B C=AB AC.5. To “keep track” of row operations on a matrix A, append an identity matrix; e.g.start withA I.After row operations, you get, say,AE.Since [A,E] is obtained by row operations, it is some matrix times [A, I]. But there is nochoice—to get E in the right part, [A, I] must be multiplied by E,and[A,E]=E[A, I].In summary, A= EA,andE tells us exactly what linear combinations of the rows of Athe rows of Aare. If, for example, A has three rows a1, a2, a3andthetoprowofE is(3, 4, 5), then the top row of Ais 3a1+4a2+5a3.6. An example of block multiplication of matrices isA BCD=AC + BD.From this, it is easy to see that the rows of [I, A] are orthogonal to the rows of [−A ,I]:I A−A I =I A−AI=−A +
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