Markov Chain Calculator

Perform the Markov Chain with:
Transition Matrix A and initial state vector B

Since |A| is a 3 x 3 matrix
and |B| is a 3 x 1 matrix,
|AB| will be a 3 x 1 matrix

P⁽¹⁾ = TP⁽⁰⁾

Green Answer Entry for Row 1, Column 1

Multiply red row 1 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₁₁|

1	2	3
4	5	6
7	8	9

|AB₁₁| = A₁₁ x B₁₁ + A₁₂ x B₂₁ + A₁₃ x B₃₁
|AB₁₁| = 1 x 9 + 2 x 6 + 3 x 3
|AB₁₁| = 9 + 12 + 9
|AB₁₁| = 30 ← Green Answer Entry

Green Answer Entry for Row 2, Column 1

Multiply red row 2 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₂₁|

1	2	3
4	5	6
7	8	9

|AB₂₁| = A₂₁ x B₁₁ + A₂₂ x B₂₁ + A₂₃ x B₃₁
|AB₂₁| = 4 x 9 + 5 x 6 + 6 x 3
|AB₂₁| = 36 + 30 + 18
|AB₂₁| = 84 ← Green Answer Entry

Green Answer Entry for Row 3, Column 1

Multiply red row 3 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₃₁|

1	2	3
4	5	6
7	8	9

138

|AB₃₁| = A₃₁ x B₁₁ + A₃₂ x B₂₁ + A₃₃ x B₃₁
|AB₃₁| = 7 x 9 + 8 x 6 + 9 x 3
|AB₃₁| = 63 + 48 + 27
|AB₃₁| = 138 ← Green Answer Entry

P⁽²⁾ = TP⁽¹⁾

Green Answer Entry for Row 1, Column 1

Multiply red row 1 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₁₁|

1	2	3
4	5	6
7	8	9

138

612

|AB₁₁| = A₁₁ x B₁₁ + A₁₂ x B₂₁ + A₁₃ x B₃₁
|AB₁₁| = 1 x 30 + 2 x 84 + 3 x 138
|AB₁₁| = 30 + 168 + 414
|AB₁₁| = 612 ← Green Answer Entry

Green Answer Entry for Row 2, Column 1

Multiply red row 2 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₂₁|

1	2	3
4	5	6
7	8	9

138

612

1368

|AB₂₁| = A₂₁ x B₁₁ + A₂₂ x B₂₁ + A₂₃ x B₃₁
|AB₂₁| = 4 x 30 + 5 x 84 + 6 x 138
|AB₂₁| = 120 + 420 + 828
|AB₂₁| = 1368 ← Green Answer Entry

Green Answer Entry for Row 3, Column 1

Multiply red row 3 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₃₁|

1	2	3
4	5	6
7	8	9

138

612

1368

2124

|AB₃₁| = A₃₁ x B₁₁ + A₃₂ x B₂₁ + A₃₃ x B₃₁
|AB₃₁| = 7 x 30 + 8 x 84 + 9 x 138
|AB₃₁| = 210 + 672 + 1242
|AB₃₁| = 2124 ← Green Answer Entry

P⁽³⁾ = TP⁽²⁾

Green Answer Entry for Row 1, Column 1

Multiply red row 1 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₁₁|

1	2	3
4	5	6
7	8	9

612

1368

2124

9720

|AB₁₁| = A₁₁ x B₁₁ + A₁₂ x B₂₁ + A₁₃ x B₃₁
|AB₁₁| = 1 x 612 + 2 x 1368 + 3 x 2124
|AB₁₁| = 612 + 2736 + 6372
|AB₁₁| = 9720 ← Green Answer Entry

Green Answer Entry for Row 2, Column 1

Multiply red row 2 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₂₁|

1	2	3
4	5	6
7	8	9

612

1368

2124

9720

22032

|AB₂₁| = A₂₁ x B₁₁ + A₂₂ x B₂₁ + A₂₃ x B₃₁
|AB₂₁| = 4 x 612 + 5 x 1368 + 6 x 2124
|AB₂₁| = 2448 + 6840 + 12744
|AB₂₁| = 22032 ← Green Answer Entry

Green Answer Entry for Row 3, Column 1

Multiply red row 3 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₃₁|

1	2	3
4	5	6
7	8	9

612

1368

2124

9720

22032

34344

|AB₃₁| = A₃₁ x B₁₁ + A₃₂ x B₂₁ + A₃₃ x B₃₁
|AB₃₁| = 7 x 612 + 8 x 1368 + 9 x 2124
|AB₃₁| = 4284 + 10944 + 19116
|AB₃₁| = 34344 ← Green Answer Entry

P⁽⁴⁾ = TP⁽³⁾

Green Answer Entry for Row 1, Column 1

Multiply red row 1 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₁₁|

1	2	3
4	5	6
7	8	9

9720

22032

34344

156816

|AB₁₁| = A₁₁ x B₁₁ + A₁₂ x B₂₁ + A₁₃ x B₃₁
|AB₁₁| = 1 x 9720 + 2 x 22032 + 3 x 34344
|AB₁₁| = 9720 + 44064 + 103032
|AB₁₁| = 156816 ← Green Answer Entry

Green Answer Entry for Row 2, Column 1

Multiply red row 2 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₂₁|

1	2	3
4	5	6
7	8	9

9720

22032

34344

156816

355104

|AB₂₁| = A₂₁ x B₁₁ + A₂₂ x B₂₁ + A₂₃ x B₃₁
|AB₂₁| = 4 x 9720 + 5 x 22032 + 6 x 34344
|AB₂₁| = 38880 + 110160 + 206064
|AB₂₁| = 355104 ← Green Answer Entry

Green Answer Entry for Row 3, Column 1

Multiply red row 3 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₃₁|

1	2	3
4	5	6
7	8	9

9720

22032

34344

156816

355104

553392

|AB₃₁| = A₃₁ x B₁₁ + A₃₂ x B₂₁ + A₃₃ x B₃₁
|AB₃₁| = 7 x 9720 + 8 x 22032 + 9 x 34344
|AB₃₁| = 68040 + 176256 + 309096
|AB₃₁| = 553392 ← Green Answer Entry

P⁽⁵⁾ = TP⁽⁴⁾

Green Answer Entry for Row 1, Column 1

Multiply red row 1 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₁₁|

1	2	3
4	5	6
7	8	9

156816

355104

553392

2527200

|AB₁₁| = A₁₁ x B₁₁ + A₁₂ x B₂₁ + A₁₃ x B₃₁
|AB₁₁| = 1 x 156816 + 2 x 355104 + 3 x 553392
|AB₁₁| = 156816 + 710208 + 1660176
|AB₁₁| = 2527200 ← Green Answer Entry

Green Answer Entry for Row 2, Column 1

Multiply red row 2 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₂₁|

1	2	3
4	5	6
7	8	9

156816

355104

553392

2527200

5723136

|AB₂₁| = A₂₁ x B₁₁ + A₂₂ x B₂₁ + A₂₃ x B₃₁
|AB₂₁| = 4 x 156816 + 5 x 355104 + 6 x 553392
|AB₂₁| = 627264 + 1775520 + 3320352
|AB₂₁| = 5723136 ← Green Answer Entry

Green Answer Entry for Row 3, Column 1

Multiply red row 3 entries in |A|
by the blue column 1 entries in |B|
to get answer entry |AB₃₁|

1	2	3
4	5	6
7	8	9

156816

355104

553392

2527200

5723136

8919072

|AB₃₁| = A₃₁ x B₁₁ + A₃₂ x B₂₁ + A₃₃ x B₃₁
|AB₃₁| = 7 x 156816 + 8 x 355104 + 9 x 553392
|AB₃₁| = 1097712 + 2840832 + 4980528
|AB₃₁| = 8919072 ← Green Answer Entry

How does the Markov Chain Calculator work?

Free Markov Chain Calculator - Given a transition matrix and initial state vector, this runs a Markov Chain process.
This calculator has 1 input.

What 1 formula is used for the Markov Chain Calculator?

What 5 concepts are covered in the Markov Chain Calculator?

exponent: The power to raise a number
formula: a fact or a rule written with mathematical symbols. A concise way of expressing information symbolically.
markov chain: a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event.
matrix: a rectangular array of numbers or symbols which are generally arranged in rows and columns
vector: directed line segment