Matrix
- Rectangular Array of Numbers in rows and columns
- Example - [5,2,-3]
- Order of Matrix is represented as Rows X Columns
- Row matrix (1 Row, any number of columns)
- Column matrix (1 Column, Multiple Rows)
- Diagonal matrix (Square matrix, Except diagonal elements every other elements are 0)
- Scalar matrix (Square matrix & Diagonal matrix in which all diagonal elements are same)
- Identity matrix - Denoted by I - Diagonal Elements are 1 (Square & Diagonal Matrix)
- Transpose of matrix - Matrix where rows and columns are interchanged
[ 1, 1, 1 ]
A = [ 2, 2, 2 ]
[ 3, 3, 3 ]
Transpose is
A' = [ 1, 2, 3 ]
[ 1, 2, 3 ]
[ 1, 2, 3 ]
Addition of Matrices
- They should be of same order
- Add Corresponding elements
A = [2,3,5]
[5,7,-2]
[5,3,0]
B = [7,-1,5]
[0,2,3]
[7,5,2]
Result ( Add Corresponding elements in same positions)
A + B = [9,2,10]
[5,9,1]
[12,8,2]
Subtraction (similar to addition)
- Order needs to match
A = [7,-2]
[0,3]
B = [0,2]
[3,5]
A - B = [7,-4]
[-3,-2]
Scalar Multiplication
- Multiplying constant with a Matrix
- Every Element of Matrix Multipled
c = [3,5]
[-2,-10]
-2c = [-6,-10]
[4,20]
Matrix Multiplication
- A (m x n)
- B (n x p)
- Columns in a (n) = Number of Rows in B (n)
A = [3,5]
[7,2]
[2,3]
B = [-2,5]
[3,7]
AB = Operate First Row (Operate) Multiple with First Coulmn
A = [3,5] --->
[7,2]
[2,3]
Select Column
B = [-2,5]
[3,7]
= [ 3 X -2 + 5 X 3, 3 X 5 + 5 X 7 ]
[ 7 X -2 + 2 X 3, 7 X 5 + 2 X 7 ]
[ 2 X -2 + 3 X 3, 2 X 5 + 3 X 7 ]
= [9,50]
[-8,49]
[5,31]
Matrix Multiplication is not commutative
Determinant of Square Matrix
+ -
A = [5,6]
[3,-4]
|A| = [Multiply Principal Diagonal Elements ] - [Subtract the Next Diagonal Elements]
= (5 X -4) - (3 X 6)
= -20 - 18
= -38
Determinant of 3 X 3 Matrix
+,-,+
A = [3,-2,1]
[2,3,4]
[2,5,4]
= +3 X [3,4] - (-2) X [2,4] + 1 X [2,3]
[5,4] [2,4] [2,5]
= 3(12-20) + 2 (8-8) + 1(10-6)
= -24 + 0 + 4
= -20
References - Link
Happy Learning!!!
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