This document provides an overview of matrix algebra concepts including: - Matrices are rectangular arrays of numbers that let us express large amounts of data in an organized form. Common types include vectors, square matrices, and rectangular matrices. - Matrices can be added or multiplied by scalars if they are the same size. Matrix multiplication follows specific rules such as being non-commutative. - Special types of matrices include symmetric, skew-symmetric, upper/lower triangular, diagonal, and identity matrices. - Determinants are values that can be calculated for square matrices and represent a type of scaling factor.

1. Section 1 : Linear
Algebra
Topic 1 : Matrix Algebra

2. What is Matrix?
 Matrix : a rectangular array of numbers or functions
 Matrices let us express large amounts of data and functions in an organized and concise
form
 e.g.
1 −1 0
2 0 3
,
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
, 𝑎1 𝑎2 𝑎3 ,
3
1
2
 Matrices having just a single row or column are called vectors.
 The second matrix is a square matrix, which means that rows and columns are equal in
number.
row vector
column
vector

3. General Concepts and Notations
 A = [ajk] =
𝑎11 𝑎12 ⋯ 𝑎1𝑛
𝑎21 𝑎22 … 𝑎2𝑛
⋮ ⋮ ⋱ ⋮
𝑎 𝑚1 𝑎 𝑚2 ⋯ 𝑎 𝑚𝑛
 This is an m × n matrix. Means a matrix with m rows and n columns. Remember, rows
always come first! This m × n is called size of a matrix.
 Each entry has two subscripts. The first is the row number and the second is the column
number. Thus a12 is the entry in row 1 and column 2.
 If m = n, then A is called an m × n square matrix. Then its diagonal containing the entries
a11, a22, …., ann is called the main diagonal of A.
 A matrix of any size m × n is called a rectangular matrix.

4. Addition of Matrices
 The sum of two matrices A = [ajk] and B = [bjk] of the same size is written A+B and has the
entries ajk + bjk obtained by adding the corresponding entries of A and B.
 Matrix of different sizes can’t be added.
 A =
−3 5 2
0 1 −3
& B =
1 0 2
−2 1 4
 A + B =
−3 + 1 5 + 0 2 + 2
0 − 2 1 + 1 −3 + 4
=
−2 5 4
−2 2 −1

5. Scalar Multiplication of Matrices
 The product of any m × n matrix A = [ajk] and any scalar c is written cA and is the m × n
matrix cA = [cajk]obtained by multiplying each entry of A by c.
 If A =
2.7 −1.8
0 0.9
 - A =
−2.7 1.8
0 −0.9
10
9
A =
3 −2
0 1
 0 × A =
0 0
0 0

6. Rules for Matrix Addition and Scalar
Multiplication
Addition rules : A, B, C  m × n matrices
 A + B = B + A c = constant number
 (A + B) + C = A + (B + C)
 A + 0 = A
 A + (-A) = 0
Multiplication rules:
 c(A + B) = cA + cB
 (c + k)A = cA + kA
 c(kA) = (ck)A

7. Matrix Multiplication
 A B = C
[m × n] [n × p] = [m × p]
Number of columns of A = Number of rows of B
e.g. A =
2 −1 1
0 3 2
5 0 1
B =
1 4 0
−1 2 1
0 3 −2
C = AB =
2 −1 1
0 3 2
5 0 1
1 4 0
−1 2 1
0 3 −2
=
2 × 1 + −1 × −1 + 1 × 0
=
3
0 × 4 + 3 × 2 + 2 × 3 =
3 9 −3
−3 12 −1
5 23 −2

8. Matrix Multiplication Rules
 AB ≠ BA (not commutative) A, B, C = three different matrices
 (kA)B = k(AB) = A (kB) k = scalar constant
 A(BC) = (AB)C
 (A + B)C = AC + BC
 C(A + B) = CA + CB

9. Transposition of a Matrix
 The transpose of a matrix is obtained by writing its rows as columns (or equivalently its
columns as rows).
 If A =
1 −1 0
2 0 3
, then AT =
1 2
−1 0
0 3
 Rules for Transposition:
a) (AT)T = A
b) (A + B)T = AT + BT
c) (cA)T = cAT
d) (AB)T = BTAT

10. Special Matrices
 Symmetric and Skew-symmetric Matrices
 A matrix is called symmetric matrix if AT = A
 A matrix is called skew-symmetric matrix if AT = -A
e.g. A =
2 12 20
12 1 15
2 15 3
is a symmetric matrix
B =
0 1 −3
−1 0 −2
3 2 0
is a skew-symmetric matrix

11. Special Matrices
 Upper and Lower Triangular Matrices
Upper triangular matrices are square matrices that can have nonzero entries only on and
above the main diagonal, whereas any entry below the diagonal must be zero.
e.g.
1 4 2
0 3 5
0 0 3
and
2 1
0 3
Lower triangular matrices can have nonzero entries only on and below the main diagonal.
e.g.
1 0 0
1 3 0
2 −2 4
and
6 0 0 0
3 5 0 0
1 2 −1 0
1 −3 6 4

12. Special Matrices
 Diagonal Matrices
These are square matrices that can have nonzero entries only on the main diagonal. Any entry
above or below the main diagonal must be zero.
Diagonal matrix : D =
1 0 0
0 3 0
0 0 0
Scalar matrix : S =
𝑐 0 0
0 𝑐 0
0 0 𝑐
, c = constant number (AS = SA = cA)
Unit matrix : I =
1 0 0
0 1 0
0 0 1
(If A is any square matrix of same size as I then AI = IA = A)

13. Determinant of a Matrix
 Determinant of a 2 × 2 matrix A =
𝑎11 𝑎12
𝑎21 𝑎22
is defined as:
 D = det A =
𝑎11 𝑎12
𝑎21 𝑎22
= a11a22 – a12a21
 Determinant of a 3 × 3 matrix B =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
is defined as:
 D = det B =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
= a11
𝑎11 𝑎12
𝑎21 𝑎22
- a12
𝑎21 𝑎23
𝑎31 𝑎33
+ a13
𝑎21 𝑎22
𝑎31 𝑎32