The document provides a comprehensive overview of matrix algebra, including definitions, types of matrices, operations, and concepts like matrix rank and determinants. Key topics include matrix equality, operations such as addition, multiplication, and scaling, as well as eigenvalues and eigenvectors with their applications. It also discusses elementary row operations and techniques for solving linear equations using these matrix concepts.

1. Basics of Matrix Algebra
Orated by:
Khushboo Gupta

2. AGENDA
• Definition and Nomenclature
• Matrix Equality
• Matrix Types
• Matrix Operations
• Independent vs. Dependent Vectors and Matrix Rank
• Matrix Determinant
• Matrix Inverse
• Examples
• Eigen values and vectors
• Applications

3. Definition and Nomenclature
• Rectangular array of numbers arranged in rows and columns represented generally as A, B,
or C.
• Number of rows and columns that a matrix has is called its dimension or order
• Numbers that appear in the rows and columns are its elements.
• Symbolically depicted as:
𝐴 𝟏𝟏 𝐴 𝟏𝟐 𝐴 𝟏𝟑
𝐴 𝟐𝟏 𝐴 𝟐𝟐 𝐴 𝟐𝟑
𝐴 𝟑𝟏 𝐴 𝟑𝟐 𝐴 𝟑𝟑
• First subscript refers to row number,second to column number.
• Another approach for representation: A = [ Aij ] where i = 1, 2 and j = 1, 2, 3, 4 which means 2
rows and 4 columns.
Aij
Matrix Equality
• Two matrices are equal if:
- Each matrix has same number of rows and columns.
- Corresponding elements within each matrix are equal
• Example:
A=
111 𝑥
𝑦 444
B=
111 222
333 444
C=
𝑙 𝑚 𝑛
𝑜 𝑝 𝑞
𝑟 𝑠 𝑡
• Matrix C is not equal to A or B, because C has more columns than A or B.

4. Matrix Types
1) Transpose Matrix; wherein row becomes columns of original matrix, say A, denoted as A’ or A
T
Example: A=
1 3 4
2 5 7
A’ or AT=
1
3
2
5
4 7
• Row 1 of matrix A becomes column 1 of A';
• Matrix order is reversed after being transposed as Matrix A is 2 x 3 matrix, but matrix A' is a 3 x 2 matrix.
2) Vectors; wherein there exist only one column or one row, chiefly of two types: column vectors and
row vectors.
Example: matrix A is a column vector, and matrix A' is a row vector as shown below:
A=
2
3
4
A’= 2 3 4
3) Square matrix: n x n matrix (same number of rows as columns). Several kinds of square matrix.
(i) Symmetric matrix: Transpose of a matrix is equal to itself. Example as shown below:
A= A’=
1 2
2 3
B= B’=
5 6 7
6 3 2
7 2 1
(ii) Diagonal matrix: Matrix with zeros in off-diagonal elements as shown below:
A=
5 0 0
0 5 0
0 0 5
B=
3 2
0 9
Diagonal Matrix Non-Diagonal Matrix
Also a Scalar Matrix

5. special kind of diagonal matrix with equal-valued elements along the diagonal
One more example is: C=
1 0
0 1 This is also called as Identity Matrix I
Matrix Types
(iii) Triangular matrix; again is of two types, chiefly:
a) Upper Triangular; if all the entries below the main diagonal/ principle diagonal are zero.
Eg.: U =
1 3 6
0 1 4
0 0 1
𝑈 𝟏𝟏 ⋯ 𝑈 𝟏𝒏
⋮ ⋱ ⋮
0 ⋯ 𝑈 𝒏𝒏
b) Lower Triangular; all the entries above the main diagonal are zero.
Eg.: L=
1 0 0
2 1 0
3 5 1
𝐿 𝟏𝟏 ⋯ 0
⋮ ⋱ ⋮
𝐿 𝒏𝟏 ⋯ 𝐿 𝟏𝒏
Conclusively, triangular matrix is one that is either lower triangular or upper triangular.
Generalized Eqn
Generalized Eqn

6. Other Types of Matrices
Matrix Name Description
Idempotent Matrix A matrix wherein A2=A
Involutory Matrix A matrix s.t. A2=I
Nilpotent Matrix A square matrix is nilpotent if there exists a
positive integer ‘m’ such that Am=0
Orthogonal matrix Square matrix which satisfies the condition
AT = A-1
Conjugate Matrix or Hermitian Matrix ( 𝐴) Obtained by negating r imaginary parts of A
matrix. Eg: A=
1 + 𝑖 2
3 4 − 2𝑖
Then,
𝐴 =
1 − 𝑖 2
3 4 + 2𝑖
Transposed Conjugate Matrix ( 𝐴T) Transpose conjugate matrix
( 𝐴T) =
1 − 𝑖 3
2 4 + 2𝑖

7. Matrix Operations
1) Addition and Subtraction
• Two matrices can be added or subtracted if they have same order.
• Just add or subtract corresponding elements.
• Example, consider matrix A and matrix B as below:
A=
1 2 3
7 8 9 2x3 B=
5 6 7
3 4 5 2x3
Thus, A + B =
1 + 5 2 + 6 3 + 7
7 + 3 8 + 4 9 + 5
=
6 8 10
10 12 14
And, A – B =
1 − 5 2 − 6 3 − 7
7 − 3 8 − 4 9 − 5
=
−4 − 4 −4
4 4 4
2) Scalar Multiplication: Multiply every element in the matrix by scalar number,
producing a new matrix, called a scalar multiple.
Example: if x = 5, and matrix A= 2 4
6 5
Then, x A = 5A = 5
2 4
6 5
=
10 20
30 25

8. 3) Matrix Multiplication: matrix product, say AB is defined only when number of columns in A
equals number of rows in B.
• A is i x j matrix, B is j x k matrix.
• Matrix product AB results in matrix, say C of order i x k
• Each element in C can be computed according to: Cik = Σj AijBjk
• Here, Cik = the element in row i and column k from matrix C
Aij = the element in row i and column j from matrix A
Bjk = the element in row j and column k from matrix B
Σj = summation sign, which indicates that the aijbjk terms should be summed over j.
Example: A=
0 1 2
3 4 5 2x3B=
6 7
8 9
10 11
3x2
Let AB = Cof order 2x2. Using formula Cik = Σj AijBjk, we get:
C11 = Σ A1jBj1 = 0*6 + 1*8 +2*10 = 0 + 8 + 20 = 28
C12 = Σ A1jBj2 = 0*7 + 1*9 +2*11 = 0 + 9 + 22 = 31
C21 = Σ A2jBj1 = 3*6 + 4*8 +5*10 = = 18 + 32 + 50 = 100
C22 = Σ A2jBj2 = 3*7 + 4*9 +5*11 = 21 + 36 +55 = 112
C=
28 31
100 112
Matrix Operations

9. Properties of Matrix Addition, Scalar and Matrix Multiplication

10. 4) Vector Multiplication; namely of 2 types:
(i) Vector Inner Product(Dot Product/ Scalar Product)If a and b are vectors, each with same
number of elements then, inner product of a and b is s and a'b = b'a = s
• Here, a and b are column vectors, each having n elements,
a' is transpose of a, which makes a' a row vector,
b' is transpose of b, which makes b' a row vector, and
s is a scalar; that is, s is a real number (not a matrix).
• Example: a=
1
2
3
b=
4
5
6
(ii) Vector Outer Product If a and b are vectors and C is their outer product shown as: ab'= C
- Here, a is a column vector, having m elements,
b is a column vector, having n elements,
b' is the transpose of b, which makes b' a row vector, and
C is a rectangular m x n matrix
• Unlike the inner product, outer product of two vectors produces a rectangular matrix, not a scalar.
• Example: a=
𝑣
𝑤
b=
𝑥
𝑦
𝑧
Matrix Operations
a'b = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32
C= ab’=
𝑣 ∗ 𝑥 𝑣 ∗ 𝑦 𝑣 ∗ 𝑧
𝑤 ∗ 𝑥 𝑤 ∗ 𝑦 𝑤 ∗ 𝑧

11. • Crucial in many matrix algebra applications, such as finding inverse of a matrix and solving
simultaneous linear equations.
• Three kinds of elementary matrix operations.
- Interchange two rows (columns).
- Multiply each element in a row (column) by a non-zero number.
- Multiply a row (column) by non-zero number, add the result to another row (or column).
• Operations on rows (columns): elementary row operations (elementary column operations).
• Elementary Operation Notation; shown below:
Elementary Matrix Operations
Operation Description Notation
Row Operations
1. Interchange rows i and j Ri <-->Rj
2. Multiply row i by s, where s ≠ 0 sRi -->Ri
3. Add s times row i to row j sRi + Rj -->Rj
Column
Operations
1. Interchange columns i and j Ci <-->Cj
2. Multiply column i by s, where s ≠ 0 sCi --> Ci
3. Add s times column i to column j sCi + Cj -->Cj

12. Elementary Matrix Operations
This introduced the concept of: Row Echelon Form (REF) and Reduced Row Echelon Form (RREF)
• Row echelon form if:
- First non-zero element in each row, called the leading entry, is 1.
- Each leading entry is in a column to the right of the leading entry in the previous row.
- Rows with all zero elements, if any, are below rows having a non-zero element
Example: Aref =
1 2 3 4
0 0 1 3
0 0 0 1
Bref =
1 2
0 1
0 0
- Row operations used to convert A into Aref is called as Gaussian Elimination
• Reduced row echelon form if:
- Matrix satisfies conditions for a row echelon form.
- Leading entry in each row is the only non-zero entry in its column.
Example: Arref =
1 2 0 0
0 0 1 0
0 0 0 1
- Row operations used to convert A (Aref) into Arref is called as Gauss–Jordan elimination

13. An Example to solve RREF (REF) using Elementary Row Transformations

14. • Vector is dependent on other, if it is a linear combination of other vectors, i.e. equal to sum of scalar
multiples of other vectors
• Suppose a = 2b + 3c, as shown: a = 2
1
2
+ 3
3
4
a =
11
16
• Need is to find Rank of a matrix
• Defined as: (a) maximum number of linearly independent column vectors in matrix
(b) the maximum number of linearly independent row vectors in the matrix.
• For an r x c matrix,
- If r is less than c, then maximum rank of the matrix is r.
- If r is greater than c, then the maximum rank of the matrix is c.
• For Null matrix rank equals zero only.
To Find Rank
• Maximum number of linearly independent vectors in a matrix equals to number of non-zero rows in
its row echelon matrix.
• Now transform matrix into REF, count number of non-zero rows.
• Consider matrix A and its Aref as shown: A =
0 1 2
1 2 1
2 7 8
Aref=
1 2 1
0 1 2
0 0 0
• Since, Aref has two non-zero rows, i.e., two independent row vector, thus, A has rank equals 2
Independent vs. Dependent Vectors and Matrix Rank
Matrix Rank
Null Matrix
O =
0 0
0 0

15. Matrix Determinant
• Only of square matrix
• Notation: Det|A| or |A| |A|=
• Determinant Computation: A is a 2 x 2 matrix above, with elements Aij, then:






2221
1211
aa
aa
21122211 aaaaA 
Properties
1 |A|=|A'|.
2 If a row or column of A = 0, then |A|= 0.
3 If every value in a row or column is multiplied by k, then |A| = k|A|.
4 If two rows or columns are identical, |A| = 0.
5 |A| remains unchanged if each element of a row or each element
multiplied by a constant, is added to any other row.
6 |AB| = |A| |B|
7 Det of a diagonal matrix = product of the diagonal elements

16. Inverse of a Matrix (A-1)
• For an n  n matrix A, there may be a B such that AB = I = BA.
• Analogous to a reciprocal
• A matrix which has an inverse (no inverse) is nonsingular (singular).
• An inverse exists only if
Properties of inverse matrices
Computation
If and |A|  0
0A
  111 --
ABAB 

   '11 -
AA' 

  AA 
11-







dc
ba
A









ac
bd-
)det(
11
A
A
Generalized formula:
A-1 = 1
𝐷𝑒𝑡|𝑨| (adjoint(A))

17. Example to solve Inverse of matrix A
Given matrix A =
3 0 2
2 0 −2
0 1 1
Step 1: Calculate matrix Minor as shown below:
Step 2: Find Cofactor Matrix as shown
=

18. Step 3: Find Adjoint (Adjugate) of matrix by transposing the matrix obtained in step 2
as shown
Step 4: Find Determinant as
Step 5: Multiply the Adj by 1/Determinant as:
Example to solve Inverse of matrix A
Thus Det|A|= 3×2 - 0×2 + 2×2 = 10
Adjoint represented as ‘adj’

19. Eigen values and vectors
• Represented as A·v=λ·v, where, A is n-by-n matrix, v is a non-zero n-by-1
vector and λ is a scalar (which may be either real or complex).
• Any value of λ for which equation above has a solution is known as an
eigenvalue of A.
• Vector, v corresponding to this value is an eigenvector.
• Rewriting eigenvalue problem as:
• Above equation is called characteristic eqn with nth order polynomial in λ with
n roots called eigenvalues of A.
A·v-λ·v=0 or A·v-λ·I·v=0 or (A-λ·I)·v=0
If v is non-zero, this equation will only have a solution if |A-λ·I|=0
EXAMPLE
Characteristic Eqn
Eigen values

20. Eigen values and vectors Contd…
• For 2 eigen values λ1 and λ2 , there will be two eigen vectors, say v1 and v2, respectively.
• Solving further for λ1 as
Thus,
Similarly taking λ2, solve for v2 as

21. Applications
• Solving system of Linear Equations
• Example: x + y + z = 62
y + 5z = -42
x + 5y - z = 27
• Representing in Matrix Form as:
• Above depiction takes an equation form as: AX = B X = A-1B
• Solving for A-1
• Computing for values of x, y, z:
Solve
A
X B

22. REFERENCES
• Robbiano, Lorenzo. "Operations with Matrices." Linear algebra.
Springer Milan, 2011. 25-48.
• Marcus, Marvin. "Linear operations on matrices." The American
Mathematical Monthly 69.9 (1962): 837-847.
• Gantmakher, Feliks Ruvimovich. The theory of matrices. Vol. 131.
American Mathematical Soc., 1998.
• Leon, Steven J. Linear algebra with applications. New York:
Macmillan, 1980.
• Jang, Jyh-Shing Roger, Chuen-Tsai Sun, and Eiji Mizutani. "Neuro-
fuzzy and soft computing, a computational approach to learning and
machine intelligence." (1997).

23. THANK YOU