4 replace a row by itself plus a multiple of another row Given \u03b1 R and i 6 j

4 replace a row by itself plus a multiple of another

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(4) (replace a row by itself plus a multiple of another row) GivenαRandi6=j, show that the matrix(I+αEij)Ais the same as the matrixAexcept with theith row replaced by itself plusαtimes thejth rowofA.
3. BLOCK MATRIX MULTIPLICATION112.2. Associativity of matrix multiplication.Note that the definition of matrix multiplication tells us thatthis operation is associative. That is, ifARm×n,BRn×k, andCRk×s, thenABRm×kso that (AB)Ciswell defined andBCRn×sso thatA(BC) is well defined, and, moreover,(AB)C=(AB)C·1(AB)C·2· · ·(AB)C·s(6)where for each= 1,2, . . . , s(AB)C·=AB·1AB·2AB·3· · ·AB·kC·=C1AB·1+C2AB·2+· · ·+Ck‘AB·k=A C1B·1+C2B·2+· · ·+Ck‘B·k=A(BC·).Therefore, we may write (6) as(AB)C=(AB)C·1(AB)C·2· · ·(AB)C·s=A(BC·1)A(BC·2). . .A(BC·s)=A BC·1BC·2. . .BC·s=A(BC).Due to this associativity property, we may dispense with the parentheses and simply writeABCfor this triplematrix product. Obviously longer products are possible.Exercise2.5.Consider the following matrices:A=23110-3B=4-10-7C=-23211-3210D=23108-5F=211210-4030-205111G=231-210-30.Using these matrices, which pairs can be multiplied together and in what order?Which triples can be multipliedtogether and in what order (e.g.the triple productBACis well defined)?Which quadruples can be multipliedtogether and in what order? Perform all of these multiplications.3. Block Matrix MultiplicationTo illustrate the general idea of block structures consider the following matrix.A=3-4110002201010-1001000214000103.Visual inspection tells us that this matrix has structure. But what is it, and how can it be represented? We re-writethe the matrix given aboveblockingout some key structures:A=3-4110002201010-1001000214000103=BI3×302×3C,whereB=3-4102210-1,C=214103,
122. REVIEW OF MATRICES AND BLOCK STRUCTURESI3×3is the 3×3 identity matrix, and 02×3is the 2×3 zero matrix.Having established this structure for thematrixA, it can now be exploited in various ways. As a simple example, we consider how it can be used in matrixmultiplication.Consider the matrixM=1204-1-12-143-20.The matrix productAMis well defined sinceAis 5×6 andMis 6×2. We show how to compute this matrix productusing the structure ofA. To do this we must firstblock decomposeMconformally with the block decomposition ofA.

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