1. Linear Algebra
Chapter 2 Matrices

2. 2.1 Addition, Scalar Multiplication, and Multiplication of Matrices
aij: the element of matrix A in row i and column j.
For a square nn matrix A, the main diagonal is:
Definition
Two matrices are equal if they are of the same size and if their corresponding elements are equal.
( for every, for all)
Thus A = B if aij = bij  i, j.

3. Addition of Matrices Definition
Let A and B be matrices of the same size.
Their sum A + B is the matrix obtained by adding together the corresponding elements of A and B.
The matrix A + B will be of the same size as A and B.
If A and B are not of the same size, they cannot be added, and we say that the sum does not exist.

4. Example 2 Determine A + B and A + C, if the sum exist. Solution
(2) Because A is 2  3 matrix and C is a 2  2 matrix, there are not of the same size, A + C does not exist.

5. Scalar Multiplication of matrices
Definition
Let A be a matrix and c be a scalar. The scalar multiple of A by c, denoted cA, is the matrix obtained by multiplying every element of A by c. The matrix cA will be the same size as A.
Example 3
Observe that A and 3A are both 2  3 matrices.

6. Negation and Subtraction
Definition
We now define subtraction of matrices in such a way that makes it compatible with addition, scalar multiplication, and negative. Let
A – B = A + (–1)B
Example

7. Multiplication of Matrices
Definition
Let the number of columns in a matrix A be the same as the number of rows in a matrix B. The product AB then exists.
Let A: mn matrix, B: nk matrix,
The product matrix C=AB has elements
C is a mk matrix.
If the number of columns in A does not equal the number of row B, we say that the product does not exist.

8. Example 4
Sol.
BA and AC do not exist.
Note. In general, ABBA.

9. Example 5
Determine AB.
Example 6
Let C = AB,
Determine c23.

10. Size of a Product Matrix
If A is an m  r matrix and B is an r  n matrix, then AB will be an m  n matrix. A
m  r B
= AB
r  n
m  n
Example 7
If A is a 5  6 matrix and B is an 6  7 matrix.
Because A has six columns and B has six rows. Thus AB exits.
And AB will be a 5  7 matrix.

11. Special Matrices Definition
A zero matrix is a matrix in which all the elements are zeros.
A diagonal matrix is a square matrix in which all the elements not on the main diagonal are zeros.
An identity matrix is a diagonal matrix in which every diagonal element is 1.

12. Theorem 2.1
Let A be m  n matrix and Omn be the zero m  n matrix. Let B be an n  n square matrix. On and In be the zero and identity n  n matrices. Then
A + Omn = Omn + A = A
BOn = OnB = On
BIn = InB = B
Example 8

13. Homework
Exercise 11
Let A be a matrix whose third row is all zeros. Let B be any matrix such that the product AB exists. Prove that the third row of AB is all zeros.
Solution

14. 2.2 Algebraic Properties of Matrix Operations
Theorem
Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed.
Properties of Matrix Addition and scalar Multiplication
1. A + B = B + A Commutative property of addition
2. A + (B + C) = (A + B) + C Associative property of addition
3. A + O = O + A = A (where O is the appropriate zero matrix)
4. c(A + B) = cA + cB Distributive property of addition
5. (a + b)C = aC + bC Distributive property of addition
6. (ab)C = a(bC)

15. Theorem
Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed.
Properties of Matrix Multiplication
1. A(BC) = (AB)C Associative property of multiplication
2. A(B + C) = AB + AC Distributive property of multiplication
3. (A + B)C = AC + BC Distributive property of multiplication
4. AIn = InA = A (where In is the appropriate zero matrix)
5. c(AB) = (cA)B = A(cB)
Note: AB BA in general. Multiplication of matrices is not commutative.

16. Proof of Thm 2.2 (A+B=B+A)
Consider the (i,j)th elements of matrices A+B and B+A:
 A+B=B+A
Example 1

17. Example 2 and 3 Compute ABC. Sol. (1) (AB)C Which method is better?
Count the number of multiplications.
26+32 =12+6=18
32+22 =6+4=10
 A(BC) is better.

18. Arithmetic Operations
If A is an m  r matrix and B is r  n matrix, the number of scalar multiplications involved in computing the product AB is mrn.
Proof
Consider the (i,j)th element of matrix AB:
AB mn  mrn
A B C = D 2   1
(1) (AB)C: 223+ 231=18
(2) A(BC): 231+ 221=10  better

19. Caution
In algebra we know that the following cancellation laws apply.
If ab = ac and a  0 then b = c.
If pq = 0 then p = 0 or q = 0.
However the corresponding results are not true for matrices.
AB = AC does not imply that B = C.
PQ = O does not imply that P = O or Q = O.
Example

20. Powers of Matrices Definition If A is a square matrix, then
Theorem 2.3
If A is an n  n square matrix and r and s are nonnegative integers, then
1. ArAs = Ar+s.
2. (Ar)s = Ars.
3. A0 = In (by definition)

21. Example 4 Solution Example 5 Simplify the following matrix expression.

22. Systems of Linear Equations
A system of m linear equations in n variables as follows
Let
We can write the system of equations in the matrix form
AX = B

23. Idempotent and Nilpotent Matrices (Exercise 30~36)
Definition
A square matrix A is said to be idempotent if A2=A.
A square matrix A is said to nilpotent if there is a positive integer p such that Ap=0. The least integer p such that Ap=0 is called the degree of nilpotency of the matrix.
Example

24. Homework Exercise 2.2: 6, 8, 11, 19, 27, 31, 33, 34, 35, 38 Exercise 8
A : m  r matrix, B: r  n matrix, C: n  s matrix. Derive formulas for the number of scalar multiplications involved in computing (AB)C and A(BC).
Exercise 27
A, B: diagonal matrix of the same size, c: scalar Prove that A+B and cA are diagonal matrices.

25. Homework Exercise 31 Determine b, c, and d such that is idempotent.
Prove that if A and B are idempotent and AB=BA, then AB is idempotent.

26. 2.3 Symmetric Matrices Definition
The transpose of a matrix A, denoted At, is the matrix whose columns are the rows of the given matrix A.
Example 1

27. Theorem 2.4 Properties of Transpose
Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed.
1. (A + B)t = At + Bt Transpose of a sum
2. (cA)t = cAt Transpose of a scalar multiple
3. (AB)t = BtAt Transpose of a product
4. (At)t = A

28. Theorem 2.4 Properties of Transpose
Proof for 3. (AB)t = BtAt

29. Symmetric Matrix Definition
A symmetric matrix is a matrix that is equal to its transpose.
match
Example
match

30. If and only if
Let p and q be statements. Suppose that p implies q (if p then q), written p  q, and that also q  p, we say that “p if and only if q” (iff )

31. Example 3
Let A and B be symmetric matrices of the same size. Prove that the product AB is symmetric if and only if AB = BA.
Proof
*We have to show (a) AB is symmetric  AB = BA, and the converse, (b) AB is symmetric  AB = BA.
() Let AB be symmetric, then
AB= (AB)t by definition of symmetric matrix
= BtAt by Thm 2.4 (3)
= BA since A and B are symmetric
() Let AB = BA, then
(AB)t = (BA)t
= AtBt by Thm 2.4 (3)
= AB since A and B are symmetric

32. Example 3 Let A be a symmetric matrix. Prove that A2 is symmetric.
Proof

33. Definition
Let A be a square matrix. The trace of A, denoted tr(A) is the sum of the diagonal elements of A. Thus if A is an n  n matrix.
tr(A) = a11 + a22 + … + ann
Example 4
Determine the trace of the matrix
Solution
We get

34. Theorem 2.5 Properties of Trace
Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed.
1. tr(A + B) = tr(A) + tr(B)
2. tr(AB) = tr(BA)
3. tr(cA) = c tr (A)
4. tr(At) = tr(A)
Proof of (1)
Since the diagonal element of A + B are (a11+b11), (a22+b22), …, (ann+bnn), we get
tr(A + B) = (a11 + b11) + (a22 + b22) + …+ (ann + bnn)
= (a11 + a22 + … + ann) + (b11 + b22 + … + bnn)
= tr(A) + tr(B).

35. Example of (2) tr(AB)=tr(BA)

36. Matrices with Complex Elements
The element of a matrix may be complex numbers. A complex number is of the form
z = a + bi
Where a and b are real numbers and a is called the real part and b the imaginary part of z.
The conjugate of a complex number z = a + bi is z = a - bi

37. Example 6
Compute A + B, 2A, and AB.
Solution

38. Definition
(i) The conjugate of a matrix A is denoted A and is obtained by taking the conjugate of each element of the matrix.
(ii) The conjugate transpose of A is written and defined by A*=A t.
(iii) A square matrix C is said to be hermitian if C=C*.
Example (i), (ii)
Example (iii)

39. Homework Exercise 2.3: 2, 5, 11, 13, 15, 23 Exercise 11
A matrix A is said to be antisymmetric if A = -At. (a) give an example of an antisymmetric matrix. (b) Prove that an antisymmetric matrix is a square matrix having diagonal elements zero. (c) Prove that the sum of two antisymmetric matrices of the same size is an antisymmetric matrix.

40. Homework
Exercise 13
Prove that any square matrix A can be decomposed into the sum of a symmetric matrix B and an antisymmetric matrix C: A = B+C.
(Find B: B = Bt, C: C = -Ct, such that A = B + C.)
Solution
(1)
A = B + C
 A = Bt - Ct
 At = B - C
(2)
(1)+(2):
A + At = 2B
 B = ½(A + At )
(1)-(2):
A - At = 2C
 C = ½(A - At )

41. 2.4 The Inverse of a Matrix Definition
Let A be an n  n matrix. If a matrix B can be found such that AB = BA = In, then A is said to be invertible and B is called the inverse of A. If such a matrix B does not exist, then A has no inverse. (denote B = A-1, and A-k=(A-1)k )
Example 1
Prove that the matrix has inverse
Proof
Thus AB = BA = I2, proving that the matrix A has inverse B.

42. Theorem 2.7 The inverse of an invertible matrix is unique. Proof
Let B and C be inverses of A.
Thus AB = BA = In, and AC = CA = In.
Multiply both sides of the equation AB = In by C.
C(AB) = CIn
(CA)B = C
InB = C
B = C
Thus an invertible matrix has only one inverse.
Thm2.2

43. A: an invertible nn matrix How to find A-1?
We shall find A-1 by finding X1, X2, …, Xn.
Since AA-1 =In, then
Solve these systems by using Gauss-Jordan elimination:
Note. [In:B] ， A-1。

44. Gauss-Jordan Elimination for finding the Inverse of a Matrix
Let A be an n  n matrix.
1. Adjoin the identity n  n matrix In to A to form the matrix [A : In].
2. Compute the reduced echelon form of [A : In].
If the reduced echelon form is of the type [In : B], then B is the inverse of A.
If the reduced echelon form is not of the type [In : B], in that the first n  n submatrix is not In, then A has no inverse.
An n  n matrix A is invertible if and only if its reduced echelon form is In.

45. Example 2
Determine the inverse of the matrix
Solution
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46. Example 3 Determine the inverse of the following matrix, if it exist.
Solution
There is no need to proceed further.
The reduced echelon form cannot have a one in the (3, 3) location.
The reduced echelon form cannot be of the form [In : B].
Thus A–1 does not exist.

47. Properties of Matrix Inverse
Let A and B be invertible matrices and c a nonzero scalar, Then
Proof
1. By definition, AA-1=A-1A=I.

48. Example 4
Solution

49. Theorem 2.8
Let AX = Y be a system of n linear equations in n variables. If A–1 exists, the solution is unique and is given by X = A–1Y.
Proof
(X = A–1Y is a solution.) Substitute X = A–1Y into the matrix equation.
AX = A(A–1Y) = (AA–1)Y = InY = Y.
(The solution is unique.)
Let X1 be any solution, thus AX1 = Y. Multiplying both sides of this equation by A–1 gives
A–1AX1= A–1Y InX1 = A–1Y X1 = A–1Y.

50. Example 5 Solve the system of equations Solution
This system can be written in the following matrix form:
If the matrix of coefficients is invertible, the unique solution is
This inverse has already been found in Example 2. We get

51. Elementary Matrices Definition
An elementary matrix is one that can be obtained from the identity matrix In through a single elementary row operation.
Example
R2  R3
5R2
R2+ 2R1

52. Elementary Matrices elementary row operation, elementary matrix。
R2  R3
5R2
R2+ 2R1

53. Notes for elementary matrices
Each elementary matrix is invertible.
Example
If A and B are row equivalent matrices and A is invertible, then B is invertible.
Proof
If A  …  B, then
B=En … E2 E1 A for some elementary matrices En, … , E2 and E1.
So B-1 = (En … E2 E1A)-1 =A-1E1-1 E2-1 … En-1.

54. Homework Exercise 2.4: 6, 14, 15, 16, 19, 21, 27 (section 2.5~2.7)
If , show that

55. Homework Exercise 21 Prove that (AtBt)-1 = (A-1B-1)t. Exercise 27
True or False: (a) If A is invertible A-1 is invertible. (b) If A is invertible A2 is invertible. (c) If A has a zero on the main diagonal it is not invertible. (d) If A is not invertible then AB is not invertible. (e) A-1 is row equivalent to In.