Let E be the elementary matrix E₃ of Example 6. Show that EA is the result of interchanging the first two rows of the matrix A.

Example 6

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Example 6 We obtain some typical elementary matrices as follows. 1 0 0 1 0 [9] 0 1 0 0 0 1 0 1 001 0 0 (3) R₁ (2) R₁ + R3 SWAP(R₁, R₂) [3]= 1 0 2 = E₁ 0 0 1 0 0 1 0 1 0 10 0 0 01 = E₂ =E3 The three elementary matrices E₁, E2, and E3 correspond to three typical elementary row operations. Now, suppose that the mxm elementary matrix E corresponds to a certain elementary row operation. It turns out that if we perform this same operation on an arbitrary mxn matrix A, we get the product matrix EA that results upon multiplying A on the left by the matrix E. Thus we can carry out an elementary row operation by means of left multiplication by the corresponding elementary matrix. Problems 38-40 illustrate typical cases in the proof of the following theorem.