**Unformatted text preview: **×1+5×0 3×0+1×1
2×0+5×1 = 31
25 . and
I2 A = We observe that I2 A = A I2 = A. 2 The identity I2 A = A I2 = A which we observed in the last example is true for
every 2 × 2 matrix A. Moreover, an analogous property holds for the n × n identity
matrix.
Lemma 10.19 (multiplication by the identity matrix)
Let A be an m × n matrix, let B be a n × ℓ matrix, and let In be the n × n identity
matrix. Then we have
A In = A and In B = B. (10.1) In particular, if A is an n × n square matrix, then (10.1) implies A In = In A = A.
We observe that In plays the role in the multiplication of n × n matrices that the
real number 1 plays in the multiplication of real numbers, that is,
x1 = 1x = x for any real number x ∈ R. 272 10. Matrices Theorem 10.20 (associative law of matrix multiplication)
Let A be an m × n matrix, B an n × ℓ matrix, and C an ℓ × k matrix. Then
(A B ) C = A (B C ). Example 10.21 (associative law of matrix multiplication)
Verify the associative law for the three matrices
31
25 A= , 1 −1
−1
1 B= , 12
21 C= . Solution: We have have
AB = 31
25 1 −1
−1
1 = 3 − 1 −3 + 1
2 − 5 −2 + 5 12
21 = 2−4
4−2
−3 + 6 −6 + 3 2 −2
−3
3 = and
(A B ) C = 2 −2
−3
3 = −2
2
3 −3 . On the other hand
BC = 1 −1
−1
1 12
21 A (B C ) = 31
25 −1
1
1 −1 1−2
2−1
−1 + 2 −2 + 1 = = −1
1
1 −1 −3 + 1 3 − 1
−2 + 5 2 − 5 = −2
2
3 −3 and
= and we ﬁnd indeed (A B ) C = A (B C ). Deﬁnition 10.22 (k -fold product of a matrix)
For any integer k ∈ N, we denote by Ak the k -fold product of A, that is,
Ak = A A . . . A
k -times Finally we introduce the transpose of a matrix. ,
2 10. Matrices 273 Deﬁnition 10.23 (transpose of a matrix)
The transpose AT of an m × n matrix A is the n × m matrix AT obtained by
interchanging the rows and columns of A. In formulas,
[AT ]i,j = aj,i for all 1 ≤ i ≤ n and 1 ≤ j ≤ m. (10.2) Example 10.24 (transpose of a matrix) A= 231
210 22
AT = 3 1 .
10 and Remark 10.25 (transpose of transpose gives original matrix)
For any matrix A, we have
(AT )T = A.
Proof: Indeed, if A is an m × n matrix, then AT is an n × m matrix, and so (AT )T
is an m × n matrix. Furthermore, we have from (10.2) that
(AT )T 10.3 i,j = AT j,i = ai,j for all 1 ≤ i ≤ m and all 1 ≤ j ≤ n. 2 The Determinant of a Square Matrix The determinant associates a scalar with a matrix and can only be deﬁned for
square matrices. We will start by discussing the determinant for the special cases
of 2 × 2 matrices and 3 × 3 matrices. After that we will deﬁne the determinant
recursively for n × n matrices. We will learn several properties of determinants
that can be used to simplify the computation of determinants.
Deﬁnition 10.26 (determinant of 2 × 2 matrix)
The determinant det(A) of a 2 × 2 matrix
A= ab
cd is deﬁned by
det(A) = ab
cd = a d − b c. (10.3) 274 10. Matrices Example 10.27 (determinants of 2 × 2 matrices)
Compute the determinants of the 2 × 2 matrices
A= 13
24 and 25
4 10 B= . Solution: From (10.3), we have that
det(A) =
det(B ) = 13
24
25
4 10 = 1 × 4 − 3 × 2 = 4 − 6 = −2,
= 2 × 10 − 5 × 4 = 20 − 20 = 0. 2 Deﬁnition 10.28 (determinant of a 3 × 3 matrix)
The det...

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