MM1_Chapter_10

# In addition a11 b11 1 a12 b12 2 a21 b21

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Unformatted text preview: 12 34 are both 2 × 2 matrices. In addition, a1,1 = b1,1 = 1, a1,2 = b1,2 = 2, a2,1 = b2,1 = 3, Thus the two matrices are equal, that is, A = B . and a2,2 = b2,2 = 4. 2 Deﬁnition 10.5 A matrix which has the same number of rows and columns is called square or a square matrix. In formulas, an m × n matrix A = (ai,j ) is square if m = n. 266 10. Matrices Example 10.6 (square matrices) Consider the matrices A= 23 21 and B= 210 231 . The matrix A is a 2 × 2 matrix and is square. The matric B is a 2 × 3 matrix and is not square. 2 Deﬁnition 10.7 (zero matrix) The m × n matrix O which has all entries zero 0 0 ··· 0 0 ··· O= . . .. . . . .. 0 0 ··· 0 0 . . . 0 is called the m × n zero matrix. 10.2 Elementary Matrix Operations In this section we learn how to multiply a matrix by a scalar and how to add and subtract matrices. Finally, we learn how to multiply an m × n matrix by an n × ℓ matrix. Deﬁnition 10.8 (scalar multiplication of a matrix by a scalar) Let A = (ai,j ) be an m × n matrix, and let λ ∈ R be a scalar (real number). Then the scalar multiplication of A by λ is the m × n matrix λ a1,1 λ a1,2 · · · λ a1,n λ a2,1 λ a2,2 · · · λ a2,n λ A = λ (ai,j ) = (λ ai,j ) = . . . . .. . . . . . . . λ am,1 λ am,2 ··· λ am,n In shorter notation, the (i, j )th entry of λ A is [λ A]i,j = λ ai,j . In words, the matrix A is multiplied with the scalar λ by multiplying every entry of the matrix by λ. 10. Matrices 267 Example 10.9 (multiplication of a matrix by a scalar) Consider the matrix 31 . A= 25 (a) If λ = 2, then λ A is given by 2A = 2×3 2×1 2×2 2×5 = 62 4 10 . (b) If λ = 1, then λ A is given by 1A = 1×3 1×1 1×2 1×5 = 31 25 = A. = 00 00 = O. (c) If λ = 0, then λ A is given by 0A = 0×3 0×1 0×2 0×5 2 We can also add and subtract matrices A and B if they have the same size, that is, the same numbers of rows and columns. Deﬁnition 10.10 (addition of matrices) Let A = (ai,j ) be an m × n matrix, and let B = (bi,j ) Then the sum A + B is the m × n matrix deﬁned by a1,1 + b1,1 a1,2 + b1,2 a2,1 + b2,1 a2,2 + b2,2 A + B = (ai,j + bi,j ) = . . . . . . am,1 + bm,1 am,2 + bm,2 be also an m × n matrix. ··· ··· .. . a1,n + b1,n a2,n + b2,n . . . ··· am,n + bm,n . In words, to add two m × n matrices A and B , we just add the corresponding elements of the matrices, that is, [A + B ]i,j = ai,j + bi,j . Note that we only can add matrices if they have the same size, that is, the same number of rows and the same number of columns. 268 10. Matrices Remark 10.11 (substraction of matrices) The diﬀerence A − B of two m × n matrices A = (ai,j ) and B = (bi,j ) can now easily be deﬁned with the help of the scalar multiplication of a matrix with a scalar and the addition of two matrices by A − B = A + (−1) B = (ai,j ) + (−bi,j ) = (ai,j − bi,j ), or equivalently, the (i, j )th entry of A − B is [A − B ]i,j = ai,j − bi,j . Example 10.12 (addition and subtraction of matrices) (a) The sum A + B of the two 2 × 3 matrices A= 231 210 and B= 226 317 , is given by A+B = 231 210 + 226 317 2+2 3+2 1+6 2+3 1+1 0+7 = = 457 527 . (b) The diﬀerence A − B of the...
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