The document discusses matrices and their operations. It defines a matrix as a rectangular array of numbers and provides examples. It describes how to add, subtract, and multiply matrices by scalars. Matrix multiplication follows certain rules regarding the number of rows and columns. The determinant of a matrix is a single number associated with square matrices. For 2x2 matrices, the determinant is calculated as the product of diagonals subtracted by the product of off-diagonals. Larger matrices use cofactor expansion. Applications include calculating the area of a triangle using a determinant of a 3x1 matrix of vertex coordinates.

1. MATRICES Merlin Priya.A
21-EDM-04

2. MATRICES AND ITS
OPERATIONS

3. MATRICES
Matrix
Rectangular array of numbers

𝑎11 𝑎12 𝑎13 ⋯ 𝑎1𝑛
𝑎21 𝑎22 𝑎23 ⋯ 𝑎2𝑛
⋮ ⋮ ⋮ ⋱ ⋮
𝑎𝑚1 𝑎𝑚2 𝑎𝑚3 ⋯ 𝑎𝑚𝑛
𝑎𝑟𝑜𝑤,𝑐𝑜𝑙𝑢𝑚𝑛
Each entry is an element
Augmented Matrix
Two matrices combined together
Order of matrix
Dimension
Rows × columns
What is the order of
1 2 3
4 5 6
?

4. MATRIX OPERATIONS
In this section, you will:
• Add and subtract matrices.
• Multiply a scalar with a matrix.
• Multiply a matrix with a matrix.

5. 9-03 MATRIX OPERATIONS
Matrix addition and subtraction
Both matrices must have same
order
Add or subtract corresponding
elements
3 1
0 2
−4 −1
+
0 −1
−2 −3
−4 −5

6. MATRIX OPERATIONS
Scalar multiplication
Multiply a matrix with a number
Distribute
3
1 2 3
0 −1 −2

7. MATRIX OPERATIONS
Matrix multiplication
Number of columns in 1st = number of rows in 2nd
 𝑚 × 𝑛 ∙ 𝑛 × 𝑝
Order of product m × p
Order is important
NO COMMUTATIVE PROPERTY!!!!!

8. MATRIX OPERATIONS
2 −1 7
0 6 −3
0
−2
3

9. DETERMINANTS OF
MATRICES
In this section, you will:
• Find a determinant 2×2 or 3×3
matrix using shortcuts.
• Find a determinant of any
square matrix using expansion
by cofactors.

10. DETERMINANTS OF MATRICES
Determinant is a real number
associated with a square matrix
2×2
If 𝐴 =
𝑎 𝑏
𝑐 𝑑
, then
det 𝐴 = 𝐴 =
𝑎 𝑏
𝑐 𝑑
= 𝑎𝑑 − 𝑏𝑐
Down product − up product
Find
1 2
3 4

11. DETERMINANTS OF MATRICES
Otherwise
Expansion by cofactors
Sign Pattern

+ − + − ⋯
− + − + ⋯
+ − + − ⋯
− + − + ⋯
⋮ ⋮ ⋮ ⋮ ⋱
Minor
Determinant of matrix created by
crossing out a row and column
Cofactor
Minor with sign from sign pattern
Given
1 0 3
2 1 0
0 2 3
, find
Minor 𝑀13
Cofactor 𝐶13

12. 9-06 APPLICATIONS OF MATRICES
Area of triangle with vertices (x1, y1),
(x2, y2), (x3, y3)
𝐴𝑟𝑒𝑎 = ±
1
2
𝑥1 𝑦1 1
𝑥2 𝑦2 1
𝑥3 𝑦3 1
Find the area of triangle with vertices
(-3, 1), (2, 4), (5, -3)