1. MAT 322: LINEAR ALGEBRA

2. MAT 322: LINEAR ALGEBRA

3. Introduction
The study of Linear Algebra essentially begins with the Mathematical array of numbers called MATRIX.
The design of this scheme assumes that students have good knowledge of matrices and matrix algebra. It is therefore necessary to assist the students by first giving a quick revision of matrix and matrix algebra.

4. A matrix is an array of numbers
Denoted with a Capital letter
Every matrix has an order (or dimension):
that is, the number of rows  the number of columns. So, A is 2 by 3 or (2  3).

5. MATRIX EQUALITY
Two matrices are equal if and only if;
they both have the same number of rows and the same number of columns
their corresponding elements are equal
SQUARE MATRIX
A square matrix is a matrix that has the same number of rows and columns (n  n)

6. DIAGONAL MATRIX
A diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero.
IDENTITY MATRIX
An identity matrix is a diagonal matrix where the diagonal elements all equal one.

7. A single row or column of numbers denoted with bold small letters
Null Matrix
A square matrix where all elements equal zero.
VECTOR
A single row or column of numbers denoted with bold small letters
row vector
column vector

8. MAIN (PRINCIPAL) DIAGONAL The elements a11, a22, a33,
MAIN (PRINCIPAL) DIAGONAL The elements a11, a22, a33, ... constitute the main or principal diagonal of the matrix A = [aij], if it is square. Eg. TRIANGULAR MATRIX A matrix in which all the entries below or above the main diagonal are zeros are called upper and lower triangular matrices respectively. If only the main diagonal is non-zero, then it is simply triangular matrix

9. Matrix Operations Transposition Addition and Subtraction
Multiplication
Inversion

10. The transpose of A is denoted by A' or (AT). If
TRANSPOSE OF A MATRIX
The transpose of a matrix is a new matrix that is formed by interchanging the rows as columns.
The transpose of A is denoted by A' or (AT). If
EXAMPLE
Given that A = then A‘ =
Notice that the first and the last elements are always the same.

12. Addition and Subtraction
Two matrices may be added (or subtracted) iff they are the same order.
Simply add (or subtract) the corresponding elements. So, A + B = C yields. Eg.
where

13. MATRIX SCALAR MULTIPLICATION
To multiply a scalar and a matrix, simply multiply each element of the matrix by the scalar quantity e.g. If k is a scalar and

14. MATRIX MULTIPLICATION
In order to multiply two matrices say AB, they must be CONFORMABLE that is, the number of columns in A must equal the number of rows in B. So,
A  B = C
(m  n)  (n  p) = (m  p)
(m  n)  (p  n) = cannot be done
(1  n)  (n  1) = a scalar (1x1)

15. Matrix Multiplication (cont.)
Thus
where

16. Matrix Multiplication- an example
Thus
where,

17. AB does not necessarily equal BA
(BA may even be an impossible operation)
Eg.,
A  B = C
(2  3)  (3  2) = (2  2)
B  A = D
(3  2)  (2  3) = (3  3)
Matrix multiplication is Associative
A(BC) = (AB)C
Multiplication and transposition
(AB)' = B'A'

18. Note that a row matrix multiplied by a column matrix when conformable gives a scalar

19. The rank of a matrix is defined as
rank(A) = number of linearly independent rows
= the number of linearly independent columns.
A set of vectors is said to be linearly dependent if scalars c1, c2, …, cn (not all zero) can be found such that: c1a1 + c2a2 + … + cnan = 0
Example:
a = [ ] and b = [1/ ] are linearly dependent
A matrix A of dimension n  p (p < n) is of rank p. Then A has maximum possible rank and is said to be of full rank.
In general, the maximum possible rank of an n  p matrix A is min(n,p).
TRACE
The trace of a matrix is the sum of the main diagonal elements of the a square matrix

20. The Inverse of a Matrix (A-1)
For an n  n matrix A, there may be a B such that AB = I = BA.
The inverse is comparable to a reciprocal
A matrix which has an inverse is nonsingular.
A matrix which does not have an inverse is singular.
An inverse exists only if
PROPERTIES OF INVERSE

21. CALCULATION OF INVERSE OF A MATRIX
If and |A|  0
ADJOINT
To get the adjoint of a matrix;
Substitute each element by its cofactor taking note of the signs.
Transpose the resulting matrix in (i) above.

23. Cofactors
The minor of each element is called a cofactor if a +ve or –ve sign is assigned to it. The sign is determined by the position of the element. If the sum i+j of the element is even, the sign is +ve. It is –ve if i+j of the element is odd.

24. EXAMPLE

25. Ex 2: (a) Find the adjoint matrix of A
(b) Use the adjoint matrix of A to find A–1
Sol:

26. cofactor matrix of A
adjoint matrix of A
inverse matrix of A
※ The computational effort of this method to derive the inverse of a matrix is high (especially to compute the cofactor matrix for a higher-order square matrix)
※ However, for computers, it is easy. since it is not necessary to judge which row operation should be used and the only thing needed to do is to calculate determinants of matrices
Check:

27. Echelon Matrix
When the number of zeros preceding the first non-zero entry of a row increases row by row until only zero rows remain; that is if there exists non-zero entries, such a matrix is called echelon matrix or is in echelon form

28. DISTINGUISHED ELEMENTS
ECHELON MATRICES
DISTINGUISHED ELEMENTS
The first non-zero in each row. Thus in eg.1 above 1 and 4 are the distinguished elements and in eg.2 the DE are 1,1,1 in eg.3 the DE are 2,7,6).
The 3rd example above is an example of a row reduced echelon matrix. The zero matrix irrespective of the number of rows or columns is a row echelon matrix.

29. ELEMENTARY ROW OPERATIONS
Two matrices A and B are said to be row equivalence if B can be obtained from A by a finite sequence of any or all the following operations called
Elementary Row Operation;
Interchange the i-th row and the j-th row
Multiply the i-th row by a non zero scalar say k
Replace the i-th row by k times the j-th row plus the i-th row
Replace the i-th row by k’ times the j-th row plus k (non-zero) times the i-th row.

30. EXAMPLES

31. EXERCISES
Transform to Echelon form
Transform to row reduced form

32. DETERMINANT OF A MATRIX
The determinant of a matrix A is denoted by |A| (or det(A)).
Determinants exist only for square matrices.
DETERMINANT OF 2X2 MATRIX
Examples

34. Ex 3: The determinant of a square matrix of order 3
Sol:

35. Subtract these three products
Alternative way to calculate the determinant of a square matrix of order 3:
Subtract these three products
Add these three products

36. –4 6 16 Ex: Recalculate the determinant of the square matrix A in Ex 36 16
※ This method is only valid for matrices of order 3

37. Ex 4: The determinant of a square matrix of order 4

38. Sol:
※ By comparing Ex 4 with Ex 3, it is apparent that the computational effort for the determinant of 4×4 matrices is much higher than that of 3×3 matrices. In the next section, we will learn a more efficient way to calculate the determinant

39. Determinant of a Triangular Matrix
If A is an n  n triangular matrix (upper triangular, lower triangular, or diagonal), then its determinant is the product of the entries on the main diagonal. That is
※ Based on the above, it is straightforward to obtain that
※ On the next slide, only take the case of upper triangular matrices for example to prove . It is straightforward to apply the following proof for the cases of lower triangular and diagonal matrices

40. Ex 6: Find the determinants of the following triangular matrices
(b)
Sol:
(a)
|A| = (2)(–2)(1)(3) = –12
(b)
|B| = (–1)(3)(2)(4)(–2) = 48

41. 3.2 Evaluation of a Determinant Using Elementary Row Operations
The computational effort to calculate the determinant of a square matrix with a large number of n is unacceptable. In this section, we show how to reduce the computational effort by using elementary operations
Note: Elementary row operations and determinants
Let A and B be square matrices
Notes: The above three properties remains valid if elementary column operations are performed to derive column-equivalent matrices (This result will be used in Ex 5 on Slide 3.25)

42. Ex: (check the characteristics of determinants

43. NOTE: Conditions that yield a zero determinant
If A is a square matrix and any of the following conditions is true, then det(A) = 0
(a) An entire row (or an entire column) consists of zeros
(Perform the cofactor expansion along the zero row or column)
(b) Two rows (or two columns) are equal
(c) One row (or column) is a multiple of another row (or column)
(For (b) and (c), based on the mathematical induction , perform the cofactor expansion along any row or column other than these two rows or columns)
Notes: For conditions (b) or (c), you can also use elementary row or column operations to create an entire row or column of zeros and obtain the results.
※ Thus, we can conclude that a square matrix has a determinant of zero if and only if it is row- (or column-) equivalent to a matrix that has at least one row (or column) consisting entirely of zeros

44. Ex:

45. PROPERTIES OF DETERMINATES
Determinants have several mathematical properties which are useful in matrix manipulations.
|A|=|A'|.
If a row or column of A = 0, then |A|= 0.
If every value in a row or column is multiplied by k, then |A| = k|A|.
If two rows (or columns) are interchanged the sign, but not value, of |A| changes.
If two rows or columns are identical, |A| = 0.
If two rows or columns are linear combination of each other, |A| = 0
|A| remains unchanged if each element of a row or each element multiplied by a constant, is added to any other row.
8. |AB| = |A| |B|
9. Det of a diagonal matrix = product of the diagonal elements

46. Determinant of a matrix product
det(AB) = det(A) det(B)
(Verified by Ex 1 on the next slide)
Notes:
(2)
(3)
(4)
(There is an example to verify this property on Slide 3.33) (Note that this property is also valid for all rows or columns other than the second row)

47. Ex 1: The determinant of a matrix product
Find |A|, |B|, and |AB|
Sol:

48. Check:
|AB| = |A| |B|

49. Ex:
Pf:

50. Ex 2: Theorem 3.6: Determinant of a scalar multiple of a matrix
If A is an n × n matrix and c is a scalar, then
det(cA) = cn det(A)
(can be proven by repeatedly use the fact that )
Ex 2:
Sol:

51. (Determinant of an invertible matrix)
A square matrix A is invertible (nonsingular) if and only if det(A)  0
If A is invertible, then AA–1 = I. , we can have |A||A–1|=|I|. Since |I|=1, neither |A| nor |A–1| is zero
Suppose |A| is nonzero. It is aimed to prove A is invertible.
By the Gauss-Jordan elimination, we can always find a matrix B, in reduced row-echelon form, that is row-equivalent to A
1. Either B has at least one row with entire zeros, then |B|=0 and thus |A|=0 since |Ek|…|E2||E1||A|=|B|. →←
2. Or B=I, then A is row-equivalent to I, and by Theorem 2.15 (Slide 2.59), it can be concluded that A is invertible

52. Ex 3: Classifying square matrices as singular or nonsingular
Sol:
A has no inverse (it is singular)
B has inverse (it is invertible/nonsingular)

53. Ex 4: Determinant of an inverse matrix Determinant of a transpose (a)
(Based on the mathematical induction , compare the cofactor expansion along the row of A and the cofactor expansion along the column of AT)
Ex 4:
(a)
(b)
Sol:

54. (2) Ax = b has a unique solution for every n × 1 matrix b
Equivalent conditions for a nonsingular matrix:
If A is an n × n matrix, then the following statements are equivalent
(1) A is invertible
(2) Ax = b has a unique solution for every n × 1 matrix b
(3) Ax = 0 has only the trivial solution
(4) A is row-equivalent to In
(5) A can be written as the product of elementary matrices
(6) det(A)  0
※ The statements (1)-(5) are collected in Theorem 2.15, and the statement (6) is from Theorem 3.7

55. Ex 5: Which of the following system has a unique solution?
(b)

56. Sol: (a) This system does not have a unique solution (b)
This system has a unique solution

57. MATRICES AND SYSTEMS OF LINEAR EQUATIONS (CRAMERS’ RULE)

60. Use Cramer’s rule to solve the system of linear equation

84. CAYLEY-HAMILTON THEOREM

85. CHECK SLIDE 90
END

86. END

87. END

88. END

89. END

90. END

91. FINAL TEST QUESTIONS Answer all questions. Time 30mins