Page 838 - Advanced_Engineering_Mathematics o'neil

818 Answers to Selected Problems

27. Let M nm be the set of all n × m real matrices. We have an addition of matrices and a multiplication of an n × m
matrix by a real number. There is a zero matrix (all entries zero), and the matrix −A =[−a ij ] serves as an additive
inverse of A. Furthermore, for any real numbers α and β,
(α + β)A = αA + βA,
(αβ)A = α(βA), and
α(A + B) = αA + αB.

Thus, M nm has the algebraic properties of a vector space. If we take the rows of an n × m matrix and simply string
then out one written after the other, then we obtain an nm vector. Thus, there is a one-to-one matching of matrices in
nm
M nm and vectors in R . This also suggests the dimension of M nm .The nm matrices formed by setting one element
nm
equal to 1 and all others zero form a basis for M nm . Thus, M nm has dimension nm, the same as R .
Section 7.2 Elementary Row Operations
−2 1 4 2 1 0 0
⎛ ⎞ ⎛ ⎞
√ √ √ √
0 3 16 3 0 3 0
⎠
1. ⎝ 3 3 ; = ⎝ ⎠
1 −2 4 8 0 0 1
⎛ ⎞ ⎛ ⎞
40 5 −15 0 5 0
√ √ √ √
−2 + 2 13 14 + 9 13 1 0 13
⎠
3. ⎝ 6 + 5 13 ; = ⎝ ⎠
2 9 5 0 0 1

30 120 0 15
5. √ √ ; = √
−3 + 2 3 15 + 8 3 1 3
⎛ ⎞ ⎛ ⎞
−1 0 3 0 1 0 0
7. ⎝ −36 28 −20 28 ⎠ ; = ⎝ 0 0 4 ⎠
−13 3 44 9 14 1 0
9. If i
= s and i
= t,
(EA) ij = (row i of E) · (column j of A)
= (I n A) ij = A ij .
Next,
(EA) sj = ( row s of E) · column j of A)
= (row t of I n ) · (column j of A)
= A tj = B sj .
Similarly, (EA) tj = A sj = B tj .

Section 7.3 The Row Echelon Form
⎛ ⎞ ⎛ ⎞
1 0 5 1 1 0
1. A R = ⎝ 0 1 2 ⎠ ; = ⎝ 0 1 0 ⎠
0 0 0 0 0 1
1 −4 −1 0 −1 0 0 1
⎛ ⎞ ⎛ ⎞
⎜0 0 0 1⎟ ⎜ 0 0 0 1⎟
0 0 0 0 0 0 1 0
3. A R = ⎝ ⎠ ; = ⎝ ⎠
0 0 0 0 0 1 0 0
1 0 0 0 1 −3
⎛ ⎞ ⎛ ⎞
⎜0 1⎟ ⎜0 0 0 1 ⎟
0 0 ⎠ ; = ⎝ 1 0 −6 17 ⎠
5. A R = ⎝
0 0 0 1 0 0

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October 14, 2010 17:50 THM/NEIL Page-818 27410_25_Ans_p801-866