By Heinonen J., Kilpelftine T.

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Definition 2 Any m x n matrix (respectively system of equations) in which each non-zero row (respectively equation) begins with more zeros (respectively zero coefficients) than the previous row (respectively equation) is said to be in (row) echelon form. • Example 2 I~ ~ ~ ~] lo and 0 0 0 O ~ [o 0 0 0 0 0 LlO ] 0 are in echelon form whilst are not. Notice the 'staircase' effect, with long but not high steps allowed. Note, too, that a given matrix, and, hence, any echelon matrix derived from it, may have columns of zeros.

The Arithmetic of Matrices 35 • Definition 2 Let both be m x n matrices (so that they have the same shape). Their sum A EB B is the m x n matrix al n 7 bin ]. «; +bmn That is, addition is componentwise. • We use the symbol EB (rather than +) to remind us that, whilst we are not actually adding numbers, we are doing something very similar - namely, adding arrays of numbers. Example 2 [2o 4 -1]~ [4~1 -3 1 2 7 1 3 Ef7 1 0 2 -5 6 -7 :] =[ ~1 5 7 -1 -4 8 -4 :] whereas [2 1 ~ ]$[ ~ :] o 1 -3 31 -7 -1 is not defined.

Let A be (say) a 2 x 2 matrix with integer coefficients, for example A =[~ ~] and let the letters Q, b, C, . . be replaced by the numbers 1, 2, 3, .... To send the message BLIACIKPloolLFIORITHIECIUP write the message in number form 2,1211,3111,161 ... etc. The coded message then comprises the number pairs A[ 1~ J. A[~ J. A[~~ J. , that is 92,26125, 71156, 431 etc. To unscramble the coded message the recipient must find a matrix B, say, which changes 92,26125,71156,431 ... etc. back to 2,1211,3111,161 ...