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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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