1. Linear Algebra - Chapter 1 [YR2005]
Lecturers:
Heru Suhartanto, PhD,
Yova Ruldeviyani, MKom,
Schedule (at C3111):
Tuesday, 1.00 – 2.40 PM
Thursday, 1.00 – 1.50 PM
Marking scheme:
2 exams (mid 15%, final 20%): reqs = attendance 75%
4 quizes, 40%
5 assignments, 25%
Reference:
Horward Anton, Elementary Linear Algebra, 8-th Ed, John Wiley & Sons, Inc, 2000
More information (will be updated soon) at
Linear Algebra - Chapter 1 [YR2005]

2. Systems of Linear Equation and Matrices
CHAPTER 1
FASILKOM UI 05

3. Introduction ~ Matrices
Information in science and mathematics is often organized into rows and columns to form rectangular arrays.
Tables of numerical data that arise from physical observations
Example: (to solve linear equations)
Solution is obtained by performing appropriate operations on this matrix
Linear Algebra - Chapter 1 [YR2005]

4. 1.1 Introduction to Systems of Linear Equations

5. Linear Algebra - Chapter 1 [YR2005]
Linear Equations
In x y variables (straight line in the xy-plane)
where a1, a2, & b are real constants,
In n variables
where a1, …, an & b are real constants
x1, …, xn = unknowns.
Example 1 Linear Equations
The equations are linear (does not involve any products or roots of variables).
Linear Algebra - Chapter 1 [YR2005]

6. Linear Algebra - Chapter 1 [YR2005]
Linear Equations
The equations are not linear.
A solution of is a sequence of n numbers s1, s2, ..., sn Э they satisfy the equation when x1=s1, x2=s2, ..., xn=sn (solution set).
Example 2 Finding a Solution Set
1 equation and 2 unknown, set one var as the parameter (assign any value) or
1 equation and 3 unknown, set 2 vars as parameter
Linear Algebra - Chapter 1 [YR2005]

7. Linear Systems / System of Linear Equations
Is A finite set of linear equations in the vars x1, ..., xn
s1, ..., sn is called a solution if x1=s1, ..., xn=sn is a solution of every equation in the system.
Ex.
x1=1, x2=2, x3=-1 the solution
x1=1, x2=8, x3=1 is not, satisfy only the first eq.
System that has no solution : inconsistent
System that has at least one solution: consistent
Consider:
Linear Algebra - Chapter 1 [YR2005]

8. Linear Algebra - Chapter 1 [YR2005]
Linear Systems
(x,y) lies on a line if and only if the numbers x and y satisfy the equation of the line. Solution: points of intersection l1 & l2
l1 and l2 may be parallel:
no intersection, no solution
l1 and l2 may intersect
at only one point: one solution
l1 and l2 may coincide:
infinite many points of intersection,
infinitely many solutions
Linear Algebra - Chapter 1 [YR2005]

9. Linear Algebra - Chapter 1 [YR2005]
Linear Systems
In general: Every system of linear equations has either no solutions, exactly one solution, or infinitely many solutions.
An arbitrary system of m linear equations in n unknowns:
a11x1 + a12x a1nxn = b1
a21x1 + a22x a2nxn = b2
am1x1 + am2x amnxn = bm
x1, ..., xn = unknowns, a’s and b’s are constants
aij, i indicates the equation in which the coefficient occurs and j indicates which unknown it multiplies
Linear Algebra - Chapter 1 [YR2005]

10. Linear Algebra - Chapter 1 [YR2005]
Augmented Matrices
Example:
Remark: when constructing, the unknowns must be written in the same order in each equation and the constants must be on the right.
Linear Algebra - Chapter 1 [YR2005]

11. Linear Algebra - Chapter 1 [YR2005]
Augmented Matrices
Basic method of solving system linear equations
Step 1: multiply an equation through by a nonzero constant.
Step 2: interchange two equations.
Step 3: add a multiple of one equation to another.
On the augmented matrix (elementary row operations):
Step 1: multiply a row through by a nonzero constant.
Step 2: interchange two rows.
Linear Algebra - Chapter 1 [YR2005]

12. Elementary Row Operations (Example)
r2= -2r1 + r2
r3 = -3r1 + r3
Linear Algebra - Chapter 1 [YR2005]

13. Elementary Row Operations (Example)
r2 = ½ r2
r3 = -3r2 + r3
r3 = -2r3
Linear Algebra - Chapter 1 [YR2005]

14. Elementary Row Operations (Example)
r1 = r1 – r2
r1 = -11/2 r3 + r1
r2 = 7/2 r3 + r2
Solution:
Linear Algebra - Chapter 1 [YR2005]

15. 1.2 Gaussian Elimination

16. Linear Algebra - Chapter 1 [YR2005]
Echelon Forms
Reduced row-echelon form, a matrix must have the following properties:
If a row does not consist entirely of zeros the the first nonzero number in the row is a 1 = leading 1
If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix.
In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.
Each column that contains a leading 1 has zeros everywhere else.
Linear Algebra - Chapter 1 [YR2005]

17. Linear Algebra - Chapter 1 [YR2005]
Echelon Forms
A matrix that has the first three properties is said to be in row-echelon form.
Example:
Reduced row-echelon form:
Row-echelon form:
Linear Algebra - Chapter 1 [YR2005]

18. Linear Algebra - Chapter 1 [YR2005]
Elimination Methods
Step 1: Locate the leftmost non zero column
Step 2: Interchange
r2 ↔ r1.
Step 3: r1 = ½ r1.
Step 4: r3 = r3 – 2r1.
Linear Algebra - Chapter 1 [YR2005]

19. Linear Algebra - Chapter 1 [YR2005]
Elimination Methods
Step 5 : continue do all steps above until the entire matrix is in row-echelon form.
r2 = -½ r2
r3 = r3 – 5r2
r3 = 2r3
Linear Algebra - Chapter 1 [YR2005]

20. Linear Algebra - Chapter 1 [YR2005]
Elimination Methods
Step 6 : add suitable multiplies of each row to the rows above to introduce zeros above the leading 1’s.
r2 = 7/2 r3 + r2
r1 = -6r3 + r1
r1 = 5r2 + r1
Linear Algebra - Chapter 1 [YR2005]

21. Linear Algebra - Chapter 1 [YR2005]
Elimination Methods
1-5 steps produce a row-echelon form (Gaussian Elimination). Step 6 is producing a reduced row-echelon (Gauss-Jordan Elimination).
Remark: Every matrix has a unique reduced row-echelon form, no matter how the row operations are varied. Row-echelon form of matrix is not unique: different sequences of row operations can produce different row- echelon forms.
Linear Algebra - Chapter 1 [YR2005]

22. Linear Algebra - Chapter 1 [YR2005]
Back-substitution
Bring the augmented matrix into row-echelon form only and then solve the corresponding system of equations by back-substitution.
Example: [Solved by back substitution]
Linear Algebra - Chapter 1 [YR2005]

23. Linear Algebra - Chapter 1 [YR2005]
Back-Substitution
Step 3. Assign arbitrary values to the free variables [parameters], if any
Linear Algebra - Chapter 1 [YR2005]

24. Homogeneous Linear Systems
A system of linear equations is said to be homogeneous if the constant terms are all zero.
Every homogeneous sytem of linear equations is consistent, since all such systems have x1=0,x2=0,...,xn=0 as a solution [trivial solution]. Other solutions are called nontrivial solutions.
Linear Algebra - Chapter 1 [YR2005]

25. Homogeneous Linear Systems
Example: [Gauss-Jordan Elimination]
Linear Algebra - Chapter 1 [YR2005]

26. Homogeneous Linear Systems
The corresponding system of equations is
Solving for the leading variables yields
The general solution is
The trivial solution is obtained when s=t=0
Linear Algebra - Chapter 1 [YR2005]

27. Homogeneous Linear Systems
Theorem:
A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions.
Linear Algebra - Chapter 1 [YR2005]

28. 1.3 Matrices and Matrix Operations

29. Matrix Notation and Terminology
A matrix is a rectangular array of numbers with rows and columns.
The numbers in the array are called the entries in the matrix.
Examples:
The size of a matrix is described in terms of the number of rows and columns its contains.
A matrix with only one column is called a column matrix or a column vector.
A matrix with only one row is called a row matrix or a row vector.
Linear Algebra - Chapter 1 [YR2005]

30. Matrix Notation and Terminology
aij = (A)ij = the entry in row i and column j of a matrix A.
1 x n row matrix a = [a1 a2 ... an]
m x 1 column matrix
A matrix A with n rows and n columns is called a square matrix of order n. Main diagonal of A = {a11, a22, ..., ann}
Linear Algebra - Chapter 1 [YR2005]

31. Operations on Matrices
Definition:
Two matrices are defined to be equal if they have the same size and their corresponding entries are equal.
If A = [aij] and B = [bij] have the same size, then A=B if and only if (A)ij=(B)ij, or equivalently aij=bij for all i and j.
If A and B are matrices of the same size, then the sum A+B is the matrix obtained by adding the entries of B to the corresponding entries of A, and the difference A–B is the matrix obtained by subtracting the entries of B from the corresponding entries of A. Matrices of different sizes cannot be added or subtracted.
Linear Algebra - Chapter 1 [YR2005]

32. Operations on Matrices
If A = [aij] and B = [bij] have the same size, then
(A+B)ij = (A)ij + (B)ij = aij + bij and
(A-B)ij = (A)ij – (B)ij = aij - bij
Definition:
If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of the matrix A by c. The matrix cA is said to be a scalar multiple of A.
If A = [aij], then (cA)ij = c(A)ij = caij.
Linear Algebra - Chapter 1 [YR2005]

33. Operations on Matrices
Definition
If A is an mxr matrix and B is an rxn matrix, then the product AB is the mxn matrix whose entries are determined as follows. To find the entry in row i and column j of AB, single out row i from the matrix A and column j from the matrix B. Multiply the corresponding entries from the row and column together and then add up the resulting products.
Linear Algebra - Chapter 1 [YR2005]

34. Linear Algebra - Chapter 1 [YR2005]
Partitioned Matrices
Linear Algebra - Chapter 1 [YR2005]

35. Matrix Multiplication by Columns and by Rows
Linear Algebra - Chapter 1 [YR2005]

36. Matrix Products as Linear Combinations
Linear Algebra - Chapter 1 [YR2005]

37. Matrix Form of a Linear System
Linear Algebra - Chapter 1 [YR2005]

38. Linear Algebra - Chapter 1 [YR2005]
Transpose of a Matrix
Linear Algebra - Chapter 1 [YR2005]

39. 1.4 Inverses; Rules of Matrix Arithmetic

40. Properties of Matrix Operations
ab = ba for real numbers a & b, but AB ≠ BA even if both AB & BA are defined and have the same size.
Example:
Linear Algebra - Chapter 1 [YR2005]

41. Properties of Matrix Operations
Theorem: Properties of
A+B = B+A
A+(B+C) = (A+B)+C
A(BC) = (AB)C
A(B+C) = AB+AC
(B+C)A = BA+CA
A(B-C) = AB-AC
(B-C)A = BA-CA
a(B+C) = aB+aC
a(B-C) = aB-aC
Math Arithmetic
(Commutative law for addition)
(Associative law for addition)
(Associative for multiplication)
(Left distributive law)
(Right distributive law)
(a+b)C = aC+bC
(a-b)C = aC-bC
a(bC) = (ab)C
a(BC) = (aB)C
Linear Algebra - Chapter 1 [YR2005]

42. Properties of Matrix Operations
Proof (d):
Proof for both have the same size:
Let size A be r x m matrix, B & C be m x n (same size).
This makes A(B+C) an r x n matrix, follows that AB+AC is also an r x n matrix.
Proof that corresponding entries are equal:
Let A=[aij], B=[bij], C=[cij]
Need to show that [A(B+C)]ij = [AB+AC]ij for all values of i and j.
Use the definitions of matrix addition and matrix multiplication.
Linear Algebra - Chapter 1 [YR2005]

43. Properties of Matrix Operations
Remark: In general, given any sum or any product of matrices, pairs of parentheses can be inserted or deleted anywhere within the expression without affecting the end result.
Linear Algebra - Chapter 1 [YR2005]

44. Linear Algebra - Chapter 1 [YR2005]
Zero Matrices
A matrix, all of whose entries are zero, such as
A zero matrix will be denoted by 0 or 0mxn for the mxn zero matrix. 0 for zero matrix with one column.
Properties of zero matrices:
A + 0 = 0 + A = A
A – A = 0
0 – A = -A
A0 = 0; 0A = 0
Linear Algebra - Chapter 1 [YR2005]

45. Linear Algebra - Chapter 1 [YR2005]
Identity Matrices
Square matrices with 1’s on the main diagonal and 0’s off the main diagonal, such as
Notation: In = n x n identity matrix.
If A = m x n matrix, then:
AIn = A and InA = A
Linear Algebra - Chapter 1 [YR2005]

46. Linear Algebra - Chapter 1 [YR2005]
Identity Matrices
Example:
Theorem: If R is the reduced row-echelon form of an n x n matrix A, then either R has a row of zeros or R is the identity matrix In.
Linear Algebra - Chapter 1 [YR2005]

47. Linear Algebra - Chapter 1 [YR2005]
Identity Matrices
Definition: If A & B is a square matrix and same size Э AB = BA = I, then A is said to be invertible and B is called an inverse of A. If no such matrix B can be found, then A is said to be singular.
Example:
Linear Algebra - Chapter 1 [YR2005]

48. Properties of Inverses
Theorem:
If B and C are both inverses of the matrix A, then B = C.
If A is invertible, then its inverse will be denoted by the symbol A-1.
The matrix
is invertible if ad-bc ≠ 0, in which case the inverse is given by the formula
Linear Algebra - Chapter 1 [YR2005]

49. Properties of Inverses
Theorem: If A and B are invertible matrices of the same size, then AB is invertible and (AB)-1 = B-1A-1.
A product of any number of invertible matrices is invertible, and the inverse of the product is the product of the inverses in the reverse order.
Example:
Linear Algebra - Chapter 1 [YR2005]

50. Linear Algebra - Chapter 1 [YR2005]
Powers of a Matrix
If A is a square matrix, then we define the nonnegative integer powers of A to be
A0=I An = AA...A (n>0)
n factors
Moreover, if A is invertible, then we define the negative integer prowers to be A-n = (A-1)n = A-1A-1...A-1
Theorem: Laws of Exponents
If A is a square matrix, and r and s are integers, then ArAs = Ar+s = Ars
If A is an invertible matrix, then
A-1 is invertible and (A-1)-1 = A
An is invertible and (An)-1 = (A-1)n for n = 0, 1, 2, ...
For any nonzero scalar k, the matrix kA is invertible and (kA)-1 = 1/k A-1.
Linear Algebra - Chapter 1 [YR2005]

51. Linear Algebra - Chapter 1 [YR2005]
Powers of a Matrix
Example:
Linear Algebra - Chapter 1 [YR2005]

52. Polynomial Expressions Involving Matrices
If A is a square matrix, m x m, and if
is any polynomial, then we define
Example:
Linear Algebra - Chapter 1 [YR2005]

53. Properties of the Transpose
Theorem: If the sizes of the matrices are such that the stated operations can be performed, then
((A)T)T = A
(A+B)T = AT + BT and (A-B)T = AT – BT
(kA)T = kAT, where k is any scalar
(AB)T = BTAT
The transpose of a product of any number of matrices is equal to the product of their transpose in the reverse order.
Linear Algebra - Chapter 1 [YR2005]

54. Invertibility of a Transpose
Theorem: If A is an invertible matrix, then AT is also invertible and (AT)-1 = (A-1)T
Example:
Linear Algebra - Chapter 1 [YR2005]

55. Linear Algebra - Chapter 1 [YR2005]
Exercise
Show that if a square matrix A satisfies A2-3A+I=0, then A-1=3I-A
Let A be the matrix
Determine whether A is invertible, and if so, find its inverse. [Hint. Solve AX = I by equating corresponding entries on the two sides.]
Linear Algebra - Chapter 1 [YR2005]

56. 1.5 Elementary Matrices and a Method for Finding A-1

57. Linear Algebra - Chapter 1 [YR2005]
Elementary Matrices
Definition:
An n x n matrix is called an elementary matrix if it can be obtained from the n x n identity matrix In by performing a single elementary row operation.
Example:
Multiply the second row of I2 by -3.
Interchange the second and fourth rows of I4.
Add 3 times the third row of I3 to the first row.
Linear Algebra - Chapter 1 [YR2005]

58. Linear Algebra - Chapter 1 [YR2005]
Elementary Matrices
Theorem: (Row Operations by Matrix Multiplication)
If the elementary matrix E results from performing a certain row operation on Im and if A is an m x n matrix, then the product of EA is the matrix that results when this same row operation is performed on A.
Example:
EA is precisely the same matrix that results when we add 3 times the first row of A to the third row.
Linear Algebra - Chapter 1 [YR2005]

59. Linear Algebra - Chapter 1 [YR2005]
Elementary Matrices
If an elementary row operation is applied to an identity matrix I to produce an elementary matrix E, then there is a second row operation that, when applied to E, produces I back again.
Inverse operation
Row operation on I that produces E
Row operation on E that reproduces I
Multiply row i by c ≠ 0
Multiply row i by 1/c
Interchange rows i and j
Add c times row i to row j
Add –c times row i to row j
Linear Algebra - Chapter 1 [YR2005]

60. Linear Algebra - Chapter 1 [YR2005]
Elementary Matrices
Theorem: Every elementary matrix is invertible, and the inverse is also an elementary matrix.
Theorem: (Equivalent Statements)
If A is an n x n matrix, then the following statements are equivalent, that is, all true or all false.
A is invertible
Ax = 0 has only the trivial solution.
The reduced row-echelon form of A is In.
A is expressible as a product of elementary matrices.
Linear Algebra - Chapter 1 [YR2005]

61. Linear Algebra - Chapter 1 [YR2005]
Elementary Matrices
Proof:
Assume A is invertible and let x0 be any solution of Ax=0.
Let Ax=0 be the matrix form of the system
Linear Algebra - Chapter 1 [YR2005]

62. Linear Algebra - Chapter 1 [YR2005]
Elementary Matrices
Assumed that the reduced row-echelon form of A is In by a finite sequence of elementary row operations, such that:
By theorem, E1,…,En are invertible. Multiplying both sides of equation on the left we obtain:
This equation expresses A as a product of elementary matrices.
If A is a product of elementary matrices, then the matrix A is a product of invertible matrices, and hence is invertible.
Matrices that can be obtained from one another by a finite sequence of elementary row operations are said to be row equivalent.
An n x n matrix A is invertible if and only if it is row equivalent to the n x n identity matrix.
Linear Algebra - Chapter 1 [YR2005]

63. A Method for Inverting Matrices
To find the inverse of an invertible matrix, we must find a sequence of elementary row operations that reduces A to the identity and then perform this same sequence of operations on In to obtain A-1.
Example:
Adjoin the identity matrix to the right side of A, thereby producing a matrix of the form [A|I]
Apply row operations to this matrix until the left side is reduced to I, so the final matrix will have the form [I|A-1].
Linear Algebra - Chapter 1 [YR2005]

64. A Method for Inverting Matrices
Added –2 times the first row to the second and –1 times the first row to the third.
Added 2 times the second row to the third.
Multiplied the third row by –1.
Added 3 times the third row to the second and –3 times the third row to the first.
We added –2 times the second row to the first.
Linear Algebra - Chapter 1 [YR2005]

65. A Method for Inverting Matrices
Often it will not be known in advance whether a given matrix is invertible.
If elementary row operations are attempted on a matrix that is not invertible, then at some point in the computations a row of zeros will occur on the left side.
Example:
Added -2 times the first row to the second and
added the first row to the third.
Added the second row to the third.
Linear Algebra - Chapter 1 [YR2005]

66. Linear Algebra - Chapter 1 [YR2005]
Exercises
Consider the matrices
Find elementary matrices, E1, E2, E3, and E4, such that
E1A=B
E2B=A
E3A=C
E4C=A
Linear Algebra - Chapter 1 [YR2005]

67. Linear Algebra - Chapter 1 [YR2005]
Exercises
Express the matrix:
in the form A = E F G R, where E, F, G are elementary matrices, and R is in row-echelon form.
Linear Algebra - Chapter 1 [YR2005]

68. 1.6 Further Results on Systems of Equations and Invertibility

69. Linear Algebra - Chapter 1 [YR2005]
Linear Systems
Theorem:
Solving Linear Systems by Matrix Inversion:
If A is an invertible n x n matrix, then for each n x 1 matrix b, the system of equations Ax = b has exactly one solution, namely, x = A-1b.
Linear systems with a common coefficient matrix.
Ax=b1, Ax=b2, Ax=b3, ..., Ax=bk
If A is invertible, then the solutions
x1=A-1b1, x2=A-1b2, x3=A-1b3, ..., xk=A-1bk
This can be efficiently done using Gauss-Jordan Elimination on [A|b1|b2|...|bk]
Linear Algebra - Chapter 1 [YR2005]

70. Linear Algebra - Chapter 1 [YR2005]
Linear Systems
Example: (a) (b)
The solution:
(a) x1=1, x2=0, x3=1
(b) x1=2, x2=1, x3=-1
Linear Algebra - Chapter 1 [YR2005]

71. Properties of Invertible Matrices
Theorem: Let A be a square matrix.
If B is a square matrix satisfying BA=I, then B=A-1.
If B is a square matrix satisfying AB=I, then B=A-1.
Theorem: Equivalent Statements
A is invertible
Ax=0 has only the trivial solutions
The reduced row-echelon form of A is In
A is expresssible as a product of elementary matrices
Ax=b is consistent for every n x 1 matrix b
Ax=b has exactly one solution for every n x 1 matrix b
Linear Algebra - Chapter 1 [YR2005]

72. Properties of Invertible Matrices
Theorem: Let A and B be square matrices of the same size. If AB is invertible, then A and B must also be invertible.
A fundamental problem.
Let A be a fixed m x n matrix. Find all m x 1 matrices b such that the system of equations Ax=b is consistent.
Linear Algebra - Chapter 1 [YR2005]

73. Linear Algebra - Chapter 1 [YR2005]
Exercises
Solve the system by inverting the coefficient matrix.
Find condition that b’s must satisfy for the system to be consistent.
Linear Algebra - Chapter 1 [YR2005]

74. 1.7 Diagonal, Triangular, and Symmetric Matrices

75. Linear Algebra - Chapter 1 [YR2005]
Diagonal Matrices
A square matrix in which all the entries off the main diagonal are zero. Example:
A diagonal matrix is invertible if and only if all of its diagonal entries are nonzero.
Linear Algebra - Chapter 1 [YR2005]

76. Linear Algebra - Chapter 1 [YR2005]
Diagonal Matrices
Example:
Linear Algebra - Chapter 1 [YR2005]

77. Linear Algebra - Chapter 1 [YR2005]
Triangular Matrices
Lower triangular = a square matrix in which all the entries above the main diagonal are zero.
Upper triangular = a square matrix in which all the entries under the main diagonal are zero.
Triangular = a matrix that is either upper triangular or lower triangular.
Linear Algebra - Chapter 1 [YR2005]

78. Linear Algebra - Chapter 1 [YR2005]
Triangular Matrices
Theorem: (basic properties of triangular matrices)
The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular.
A triangular matrix is invertible if and only its diagonal entries are all nonzero.
The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular.
Linear Algebra - Chapter 1 [YR2005]

79. Linear Algebra - Chapter 1 [YR2005]
Triangular Matrices
Example:
The matrix A is invertible, since its diagonal entries are nonzero, but the matrix B is not.
This inverse is upper triangular.
This product is upper triangular.
Linear Algebra - Chapter 1 [YR2005]

80. Linear Algebra - Chapter 1 [YR2005]
Symmetric Matrices
A square matrix A is called symmetric if A = AT.
A matrix A = [aij] is symmetric if and only if aij=aji for all values of i and j.
Linear Algebra - Chapter 1 [YR2005]

81. Linear Algebra - Chapter 1 [YR2005]
Symmetric Matrices
Theorem: If A and B are symmetric matrices with the same size, and if k is any scalar, then
AT is symmetric
A+B and A-B are symmetric
kA is symmetric
Theorem:
If A is an invertible matrix, then A-1 is symmetric.
If A is an invertible matrix, then AAT and ATA are also invertible.
Linear Algebra - Chapter 1 [YR2005]

82. Linear Algebra - Chapter 1 [YR2005]
Exercise
Find all values of a, b, and c for which A is symmetric.
Find all values of a and b for which A and B are both not invertible.
Linear Algebra - Chapter 1 [YR2005]