MATH 1300 week 4 notes PDF

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1.7 1.7.1

Diagonal, Triangular and Symmetric Matrices Diagonal Matrices

Definition 1.71. A square matrix A = (aij ) for which every term of f the main diagonal is zero, that is, aij = 0 for i 6= j, is called a diagonal matrix.

Example 1.72. The following matrices are diagonal. 1 0 ✓ ◆ 3 0 0 4 0 A= and B = @ 0 2 0 A . 0 2 0 0 4 Theorem 1.73. The inverse of a diagonal matrix is a diagonal matrix obtained by taking the inverse of each diagonal entry (provided they are all nonzero).

Example 1.74. Let 1 2 0 0 A = @ 0 3 0 A, 0 0 5 0

so A−1

1 1/2 0 0 0 A. = @ 0 1/3 0 0 1/5 0

Theorem 1.75. If A is a diagonal matrix, An is a diagonal matrix obtained by taking the nth power of each diagonal entry, where n is a positive integer.

Example 1.76. Let 1 0 2 0 0 A = @ 0 3 0 A, 0 0 5

1 1 0 4 0 0 22 0 0 then A2 = @ 0 3 2 0 A = @ 0 9 0 A . 0 0 52 0 0 25 0

34

1.7.2

Triangular Matrices

Definition 1.77. A square matrix A = (aij ) is called upper triangular if aij = 0 for i > j. 1 0 a11 a12 a13 · · · · · · a1n B 0 a22 a23 · · · · · · a2n C C B B 0 0 a33 · · · · · · a3n C C B .. .. . . . C. A = B .. . . . . . . C B B . .. .. . C .. @ .. . . . A . . 0 0 0 · · · · · · ann It is called lower triangular if aij = 0 for i < j. 0

B B B B A=B B B @ Example 1.78. Consider 1 0 1 2 3 A = @ 0 1 4 A, 0 0 2

a11 0 0 a21 a22 0 a31 a32 a33 .. .. ... . . .. .. ... . . an1 an2 an3

B=

✓

3 0 1 3

◆

··· ··· ··· ··· ··· ··· .. . .. .

0 0 0 .. . .. . · · · · · · ann

1

C C C C C. C C A

0

1 2 1 2 and C = @ 0 0 1 A. 0 0 3

A and C are upper triangulars, while B is lower triangular.

Theorem 1.79. 1. If A is a lower triangular matrix, then AT is upper triangular. 2. If B is an upper triangular matrix, then B T is lower triangular. 3. If A and B are upper triangular matrices, then AB is upper triangular. 4. If A and B are lower triangular matrices, then AB is lower triangular. 5. Let A = (aij ) be an n ⇥ n triangular matrix, A is invertible if and only if aii 6= 0 for all 1  i  n. 6. If A is an invertible upper triangular matrix, then A−1 is also upper triangular. 7. If A is an invertible lower triangular matrix, then A−1 is also lower triangular. 35

Example 1.80. Let 1 1 0 0 A=@ 2 1 0 A 3 4 2 0

0

1 5 0 0 0 A. and B = @ 7 8 1 2 4

Then, 1 5 0 0 AB = @ 3 8 0 A 41 36 8 0

1.7.3

and A−1

1 1 0 0 1 0 A. = @ 2 11/2 2 1/2 0

Symmetric Matrices

Definition 1.81. A matrix A = (aij ) is called symmetric if AT = A. That is, A is symmetric if it is a square matrix for which aij = aj i . Remark: If matrix A is symmetric, then the element of A are symmetric with respect to the main diagonal of A. Example 0 1 A=@ 2 3

1.82. Consider 1 1 1 1 0 0 0 1 0 0 5 4 3 2 1 4 0 2 3 1 4 A , C = @ 4 3 2 1 A , and I3 = @ 0 1 0 A . 4 5 A , B = @ 6 0 0 1 3 2 1 0 0 6 1 5 6

A and I3 are symmetric, while B and C are not symmetric. Theorem 1.83. Suppose A and B are symmetric matrices, then 1. AT is symmetric. 2. A + B and A  B are symmetric. 3. cA is symmetric, for any scalar c. Example 1.84. Let 0

1 1 2 3 A=@ 2 4 5 A 3 5 6

0

1 0 1 0 5 8 A. and B = @ 1 0 8 7

Then, 1 1 1 3 A + B = @ 1 9 13 A 3 13 1 0

0

1 1 3 3 and A  B = @ 3 1 3 A . 3 3 13 36

Theorem 1.85. If A is an invertible symmetric matrix, then A−1 is symmetric.

Example 1.86. Let 1 1 2 3 A = @ 2 4 5 A, 3 5 6 0

then A−1

37

1 1 3 2 3 1 A . = @ 3 2 1 0 0

Chapter 2 Determinants 2.1

Determinants by Cofactor Expansion

Definition 2.1. Let A =

✓

a b c d

◆ , the determinant of the matrix A, denoted by det(A)

✓

a b c d

◆

or |A| is, ad  bc. Remark: We can now say that A =

is invertible if det(A) = ad  bc 6= 0, then

1 A = det(A)    a b . We can also write det(A) =  c d −1

✓

d b c a

◆

.

Definition 2.2. Let A = (aij ) be an n ⇥ n matrix. Let Mij be the (n  1) ⇥ (n  1) submatrix of A obtained by deleting the ith row and j th column of A. The determinant det(Mij ) is called the minor of aij . The cofactor Cij of aij is defined as Cij = (1)i+j det(Mij ).

Example02.3. 1 3 1 2 Let A = @ 4 5 6 A. Calculate C12 , C23 , and C31 . 7 1 2 Solution:

  4 6  det(M12) =  7 2

   = (4)(2)  (6)(7) = 34.  38

Then,

   3 1   = (3)(1)  (7)(1) = 10. det(M23) =  7 1     1 2    = (1)(6)  (5)(2) = 16. det(M31) =  5 6  C12 = (1)1+2 det(M12) = (1)(34) = 34.

C23 = (1)2+3 det(M23) = (1)(10) = 10. C31 = (1)3+1 det(M31) = (1)(16) = 16. Remark: The signs (1)i+j form a checkerboard pattern that has a 0 0 1 +  + +  + B  +  @  +  A, n = 3 B @ +  + +  +  + 

+ in the (1, 1) position. 1  + C C , n = 4.  A +

Theorem 2.4. Let A = (aij ) be an n ⇥ n matrix. Then, for each 1  i  n, det(A) = ai1 Ci1 + ai2 Ci2 + · · · + ain Cin (expansion of det(A) along the ith row); and for each 1  i  n, det(A) = a1j C1j + a2j C2j + · · · + anj Cnj (expansion of det(A) along the j th column).

Example 2.5. Evaluate

   1 2 3 4    4 2 1 3   .  3 0 0 3    2 0 2 3 

Solution: Expanding along the third row and using the checkerboard pattern for n = 4, we have    1 2 3  4        1 2 3   2 3 4   4 2  1 3       1  1 3  + (1)(3)  4 2   = (1)(3)  2 0 3   3 0  2 0 2   0 2 3     2 0 2 3  39

   2 3 4    1 3  by expanding along the first column and using the checkerWe now evaluate  2  0 2 3  board pattern for n = 3, obtaining    2 3 4         1 3   3 4   2 1 3  = (1)(2)  2 3 + (1)(2)  2 3  = (1)(2)(9) + (1)(2)(1) = 20.   0 2 3     1 2 3    1  by expanding along the third row and using the checkerSimilarly, we evaluate  4 2  2 0 2  board pattern for n = 3, obtaining    1 2 3         2 3   1 2  4 2  1     = (1)(2)(8) + (1)(2)(10) = 4. + (1)(2)    = (1)(2)  2 1  4 2     2 0 2  The value of the given determinant is (1)(3)(20) + (1)(3)(4) = 48.

Theorem 2.6. If a matrix A = (aij ) is upper (lower) triangular, then det(A) = a11a22 · · · ann ; that is, the determinant of a triangular matrix is the product of the elements on the main diagonal.

Corollary 2.7. The determinant of a diagonal matrix is the product of the entries on its main diagonal.

Example 2.8. Let 1 0 2 3 4 A = @ 0 4 5 A , 0 0 3

0

1 3 0 0 5 0 A, B=@ 2 6 8 4

det(A) = (2)(4)(3) = 24. det(B) = (3)(5)(4) = 60. det(C) = (5)(4)(6) = 120.

40

0

1 5 0 0 0 A. C=@ 0 4 0 0 6...