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**Unformatted text preview: **two 2 × 2 matrices
A= 12
34 12
34 − and B= 11
11 is given by
A−B = 11
11 = 1−1 2−1
3−1 4−1 = 01
23 , 11
11 = 1+1 2+1
3+1 4+1 = 23
45 . and the sum A + B is given by
A+B = 12
34 + (c) Let A be an arbitrary m × n matrix and let O be the m × n zero matrix. Then
A + O = O + A = A.
Indeed
[A+O]i,j = ai,j +0 = [O+A]i,j = 0+ai,j = ai,j for all 1 ≤ i ≤ m and all 1 ≤ j ≤ n. 10. Matrices 269 (d) If A is a matrix and k is a positive integer, then
A + A + . . . + A = k A. 2 k -times
Next we learn that we can multiply an m × n matrix A with and n × ℓ matrix B ,
and the result will be an m × ℓ matrix. The matrix multiplication A B is rather
more complicated than scalar multiplication or matrix addition, since matrix multiplication is not performed by multiplying corresponding entries but by taking the
scalar products of rows the matrix A and columns of the matrix B .
Deﬁnition 10.13 (multiplication of an m × n by an n × ℓ matrix)
Let A = (ai,j ) be an m × n matrix, and let B = (bj,k ) be an n × ℓ matrix. Then
the product A B of the m × n matrix A with the n × ℓ matrix B is deﬁned
to be the m × ℓ matrix A = ([A B ]i,j ) with the entries
n ai,k bk,j (i, j )th entry of A B = [A B ]i,j =
k =1 = scalar product of ith row vector of A and j th column vector of B a1,j a2,j = ai,1 , ai,2 , . . . , ai,n · . .
.
.
an,j Note that the product A B only exists if the number of columns in the ﬁrst
matrix A equals the number of rows in the second matrix B .
Note that if A and B are both n × n matrices, then both A B and B A exist but
in general A B and B A are not necessarily equal.
Example 10.14 (matrix multiplication)
The matrices A and B , given by
A= 31
25 and B= −2 6
47 . They are both 2 × 2 matrices, and we can compute both A B and B A. We ﬁnd
AB = 31
25 −2 6
47 270 10. Matrices = 3 × (−2) + 1 × 4
2 × (−2) + 5 × 4 = −2 25
16 47 3×6+1×7
2×6+5×7 , and
31
25 −2 6
47 BA =
= (−2) × 3 + 6 × 2
4×3+7×2 = 6 28
26 39 (−2) × 1 + 6 × 5
4×1+7×5 . We see that A B = B A. 2 Example 10.15 (matrix multiplication)
Consider the two matrices
A = (1 2) 1
B = 2 .
3 and Then A B does not exist, since A is a 1 × 2 matrix and B is a 3 × 1 matrix. The
product B A does exist, since B is a 3 × 1 and A is a 1 × 2 matrix. We ﬁnd 1
1×1 1×2
12
B A = 2 (1 2) = 2 × 1 2 × 2 = 2 4 .
2
3
3×1 3×2
36
Deﬁnition 10.16 (identity matrix)
The n × n identity matrix In is the n × n matrix which has ones on the leading
diagonal and zeros everywhere else, that is,
[In ]i,i = 1 for 1 ≤ i ≤ n and [In ]i,j = 0 if i = j. Explicitly the identity matrix In is In = 100
010
001
...
...
...
000 ···
···
···
..
. 0
0
0
.
.
. ··· 1 . 10. Matrices 271 Example 10.17 (identity matrix)
The 2 × 2 identity matrix and the 3 × 3 identity matrix are given by 100
10
and
I3 = 0 1 0 ,
I2 =
01
001
respectively. 2 Example 10.18 (multiplication with the identity matrix)
Let the 2 × 2 matrix A be given by
A= 31
25 . Compute the products I2 A and A I2 , where I2 is the 2 × 2 identity matrix.
Solution:
I2 A = 10
01 31
25 = 1×3+0×2
0×3+1×2 1×1+0×5
0×1+1×5 = 31
25 , 31
25 10
01 = 3×1+1×0
2...

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