# Matrix Exponentiation

• Difficulty Level : Hard
• Last Updated : 30 Jun, 2022

This is one of the most used techniques in competitive programming. Let us first consider below simple question.

What is the minimum time complexity to find n’th Fibonacci Number?
We can find n’th Fibonacci Number in O(Log n) time using Matrix Exponentiation. Refer method 4 of this for details. In this post, a general implementation of Matrix Exponentiation is discussed.

```For solving the matrix exponentiation we are assuming a
linear recurrence equation like below:

F(n) = a*F(n-1) + b*F(n-2) + c*F(n-3)   for n >= 3
. . . . . Equation (1)
where a, b and c are constants.

For this recurrence relation, it depends on three previous values.
Now we will try to represent Equation (1) in terms of the matrix.

[First Matrix] = [Second matrix] * [Third Matrix]
| F(n)   |     =   Matrix 'C'    *  | F(n-1) |
| F(n-1) |                          | F(n-2) |
| F(n-2) |                          | F(n-3) |

Dimension of the first matrix is 3 x 1 .
Dimension of the third matrix is also 3 x 1.

So the dimension of the second matrix must be 3 x 3
[For multiplication rule to be satisfied.]

Now we need to fill the Matrix 'C'.

So according to our equation.
F(n) = a*F(n-1) + b*F(n-2) + c*F(n-3)
F(n-1) = F(n-1)
F(n-2) = F(n-2)

C = [a b c
1 0 0
0 1 0]

Now the relation between matrix becomes :
[First Matrix]  [Second matrix]       [Third Matrix]
| F(n)   |  =  | a b c |  *           | F(n-1) |
| F(n-1) |     | 1 0 0 |              | F(n-2) |
| F(n-2) |     | 0 1 0 |              | F(n-3) |

Lets assume the initial values for this case :-
F(0) = 0
F(1) = 1
F(2) = 1

So, we need to get F(n) in terms of these values.

So, for n = 3 Equation (1) changes to
| F(3) |  =  | a b c |  *           | F(2) |
| F(2) |     | 1 0 0 |              | F(1) |
| F(1) |     | 0 1 0 |              | F(0) |

Now similarly for n = 4
| F(4) |  =  | a b c |  *           | F(3) |
| F(3) |     | 1 0 0 |              | F(2) |
| F(2) |     | 0 1 0 |              | F(1) |

- - - -  2 times - - -
| F(4) |  =  | a b c |  * | a b c | *       | F(2) |
| F(3) |     | 1 0 0 |    | 1 0 0 |         | F(1) |
| F(2) |     | 0 1 0 |    | 0 1 0 |         | F(0) |

So for n, the Equation (1) changes to

- - - - - - - - n -2 times - - - -  -
| F(n)   |  =  | a b c | * | a b c | * ... * | a b c | * | F(2) |
| F(n-1) |     | 1 0 0 |   | 1 0 0 |         | 1 0 0 |   | F(1) |
| F(n-2) |     | 0 1 0 |   | 0 1 0 |         | 0 1 0 |   | F(0) |

| F(n)   |  =  [ | a b c | ] ^ (n-2)   *  | F(2) |
| F(n-1) |     [ | 1 0 0 | ]              | F(1) |
| F(n-2) |     [ | 0 1 0 | ]              | F(0) |```

So we can simply multiply our Second matrix n-2 times and then multiply it with the third matrix to get the result. Multiplication can be done in (log n) time using Divide and Conquer algorithm for power (See this or this)

Let us consider the problem of finding n’th term of a series defined using below recurrence.

```n'th term,
F(n) = F(n-1) + F(n-2) + F(n-3), n >= 3
Base Cases :
F(0) = 0, F(1) = 1, F(2) = 1```

We can find n’th term using following :

```Putting a = 1, b = 1 and c = 1 in above formula

| F(n)   |  =  [ | 1 1 1 | ] ^ (n-2)   *  | F(2) |
| F(n-1) |     [ | 1 0 0 | ]              | F(1) |
| F(n-2) |     [ | 0 1 0 | ]              | F(0) |```
Recommended Practice

Below is the implementation of above idea.

## C++

 `// C++ program to find value of f(n) where f(n)``// is defined as``//    F(n) = F(n-1) + F(n-2) + F(n-3), n >= 3``// Base Cases :``//    F(0) = 0, F(1) = 1, F(2) = 1``#include``using` `namespace` `std;` `// A utility function to multiply two matrices``// a[][] and b[][].  Multiplication result is``// stored back in b[][]``void` `multiply(``int` `a, ``int` `b)``{``    ``// Creating an auxiliary matrix to store elements``    ``// of the multiplication matrix``    ``int` `mul;``    ``for` `(``int` `i = 0; i < 3; i++)``    ``{``        ``for` `(``int` `j = 0; j < 3; j++)``        ``{``            ``mul[i][j] = 0;``            ``for` `(``int` `k = 0; k < 3; k++)``                ``mul[i][j] += a[i][k]*b[k][j];``        ``}``    ``}` `    ``// storing the multiplication result in a[][]``    ``for` `(``int` `i=0; i<3; i++)``        ``for` `(``int` `j=0; j<3; j++)``            ``a[i][j] = mul[i][j];  ``// Updating our matrix``}` `// Function to compute F raise to power n-2.``int` `power(``int` `F, ``int` `n)``{``    ``int` `M = {{1,1,1}, {1,0,0}, {0,1,0}};` `    ``// Multiply it with initial values i.e with``    ``// F(0) = 0, F(1) = 1, F(2) = 1``    ``if` `(n==1)``        ``return` `F + F;` `    ``power(F, n/2);` `    ``multiply(F, F);` `    ``if` `(n%2 != 0)``        ``multiply(F, M);` `    ``// Multiply it with initial values i.e with``    ``// F(0) = 0, F(1) = 1, F(2) = 1``    ``return` `F + F ;``}` `// Return n'th term of a series defined using below``// recurrence relation.``// f(n) is defined as``//    f(n) = f(n-1) + f(n-2) + f(n-3), n>=3``// Base Cases :``//    f(0) = 0, f(1) = 1, f(2) = 1``int` `findNthTerm(``int` `n)``{` `    ``int` `F = {{1,1,1}, {1,0,0}, {0,1,0}} ;` `    ``//Base cases``    ``if``(n==0)``        ``return` `0;``    ``if``(n==1 || n==2)``        ``return` `1;` `    ``return` `power(F, n-2);``}` `// Driver code``int` `main()``{``   ``int` `n = 5;` `   ``cout << ``"F(5) is "` `<< findNthTerm(n);` `   ``return` `0;``}`

## Java

 `// JAVA program to find value of f(n) where``// f(n) is defined as``// F(n) = F(n-1) + F(n-2) + F(n-3), n >= 3``// Base Cases :``// F(0) = 0, F(1) = 1, F(2) = 1``import` `java.io.*;` `class` `GFG {``    ` `    ``// A utility function to multiply two``    ``// matrices a[][] and b[][].``    ``// Multiplication result is``    ``// stored back in b[][]``    ``static` `void` `multiply(``int` `a[][], ``int` `b[][])``    ``{``        ``// Creating an auxiliary matrix to``        ``// store elements of the``        ``// multiplication matrix``        ``int` `mul[][] = ``new` `int``[``3``][``3``];``        ``for` `(``int` `i = ``0``; i < ``3``; i++)``        ``{``            ``for` `(``int` `j = ``0``; j < ``3``; j++)``            ``{``                ``mul[i][j] = ``0``;``                ``for` `(``int` `k = ``0``; k < ``3``; k++)``                    ``mul[i][j] += a[i][k]``                                ``* b[k][j];``            ``}``        ``}``    ` `        ``// storing the multiplication``        ``// result in a[][]``        ``for` `(``int` `i=``0``; i<``3``; i++)``            ``for` `(``int` `j=``0``; j<``3``; j++)``            ` `                ``// Updating our matrix``                ``a[i][j] = mul[i][j];``    ``}``    ` `    ``// Function to compute F raise to``    ``// power n-2.``    ``static` `int` `power(``int` `F[][], ``int` `n)``    ``{``        ``int` `M[][] = {{``1``, ``1``, ``1``}, {``1``, ``0``, ``0``},``                               ``{``0``, ``1``, ``0``}};``    ` `        ``// Multiply it with initial values``        ``// i.e with F(0) = 0, F(1) = 1,``        ``// F(2) = 1``        ``if` `(n == ``1``)``            ``return` `F[``0``][``0``] + F[``0``][``1``];``    ` `        ``power(F, n / ``2``);``    ` `        ``multiply(F, F);``    ` `        ``if` `(n%``2` `!= ``0``)``            ``multiply(F, M);``    ` `        ``// Multiply it with initial values``        ``// i.e with F(0) = 0, F(1) = 1,``        ``// F(2) = 1``        ``return` `F[``0``][``0``] + F[``0``][``1``] ;``    ``}``    ` `    ``// Return n'th term of a series defined``    ``// using below recurrence relation.``    ``// f(n) is defined as``    ``// f(n) = f(n-1) + f(n-2) + f(n-3), n>=3``    ``// Base Cases :``    ``// f(0) = 0, f(1) = 1, f(2) = 1``    ``static` `int` `findNthTerm(``int` `n)``    ``{``        ``int` `F[][] = {{``1``, ``1``, ``1``}, {``1``, ``0``, ``0``},``                                  ``{``0``, ``1``, ``0``}} ;``    ` `        ``return` `power(F, n-``2``);``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `main (String[] args) {``        ` `        ``int` `n = ``5``;``        ` `        ``System.out.println(``"F(5) is "``                           ``+ findNthTerm(n));``    ``}``}` `//This code is contributed by vt_m.`

## Python3

 `# Python3 program to find value of f(n)``# where f(n) is defined as``# F(n) = F(n-1) + F(n-2) + F(n-3), n >= 3``# Base Cases :``# F(0) = 0, F(1) = 1, F(2) = 1` `# A utility function to multiply two``# matrices a[][] and b[][]. Multiplication``# result is stored back in b[][]``def` `multiply(a, b):``    ` `    ``# Creating an auxiliary matrix``    ``# to store elements of the``    ``# multiplication matrix``    ``mul ``=` `[[``0` `for` `x ``in` `range``(``3``)]``              ``for` `y ``in` `range``(``3``)];``    ``for` `i ``in` `range``(``3``):``        ``for` `j ``in` `range``(``3``):``            ``mul[i][j] ``=` `0``;``            ``for` `k ``in` `range``(``3``):``                ``mul[i][j] ``+``=` `a[i][k] ``*` `b[k][j];` `    ``# storing the multiplication``    ``# result in a[][]``    ``for` `i ``in` `range``(``3``):``        ``for` `j ``in` `range``(``3``):``            ``a[i][j] ``=` `mul[i][j]; ``# Updating our matrix``    ``return` `a;` `# Function to compute F raise``# to power n-2.``def` `power(F, n):` `    ``M ``=` `[[``1``, ``1``, ``1``], [``1``, ``0``, ``0``], [``0``, ``1``, ``0``]];` `    ``# Multiply it with initial values i.e``    ``# with F(0) = 0, F(1) = 1, F(2) = 1``    ``if` `(n ``=``=` `1``):``        ``return` `F[``0``][``0``] ``+` `F[``0``][``1``];` `    ``power(F, ``int``(n ``/` `2``));` `    ``F ``=` `multiply(F, F);` `    ``if` `(n ``%` `2` `!``=` `0``):``        ``F ``=` `multiply(F, M);` `    ``# Multiply it with initial values i.e``    ``# with F(0) = 0, F(1) = 1, F(2) = 1``    ``return` `F[``0``][``0``] ``+` `F[``0``][``1``] ;` `# Return n'th term of a series defined``# using below recurrence relation.``# f(n) is defined as``# f(n) = f(n-1) + f(n-2) + f(n-3), n>=3``# Base Cases :``# f(0) = 0, f(1) = 1, f(2) = 1``def` `findNthTerm(n):``    ``F ``=` `[[``1``, ``1``, ``1``], [``1``, ``0``, ``0``], [``0``, ``1``, ``0``]];` `    ``return` `power(F, n ``-` `2``);` `# Driver code``n ``=` `5``;` `print``(``"F(5) is"``,``      ``findNthTerm(n));` `# This code is contributed by mits`

## C#

 `// C# program to find value of f(n) where``// f(n) is defined as``// F(n) = F(n-1) + F(n-2) + F(n-3), n >= 3``// Base Cases :``// F(0) = 0, F(1) = 1, F(2) = 1``using` `System;` `class` `GFG {` `    ``// A utility function to multiply two``    ``// matrices a[][] and b[][]. Multiplication``    ``// result is stored back in b[][]``    ``static` `void` `multiply(``int``[, ] a, ``int``[, ] b)``    ``{``        ` `        ``// Creating an auxiliary matrix to store``        ``// elements of the multiplication matrix``        ``int``[, ] mul = ``new` `int``[3, 3];``        ` `        ``for` `(``int` `i = 0; i < 3; i++) {``            ``for` `(``int` `j = 0; j < 3; j++) {``                ``mul[i, j] = 0;``                ``for` `(``int` `k = 0; k < 3; k++)``                    ``mul[i, j] += a[i, k] * b[k, j];``            ``}``        ``}` `        ``// storing the multiplication result``        ``// in a[][]``        ``for` `(``int` `i = 0; i < 3; i++)``            ``for` `(``int` `j = 0; j < 3; j++)``            ` `                ``// Updating our matrix``                ``a[i, j] = mul[i, j];``    ``}` `    ``// Function to compute F raise to power n-2.``    ``static` `int` `power(``int``[, ] F, ``int` `n)``    ``{``        ` `        ``int``[, ] M = { { 1, 1, 1 }, { 1, 0, 0 },``                                   ``{ 0, 1, 0 } };` `        ``// Multiply it with initial values i.e``        ``// with F(0) = 0, F(1) = 1, F(2) = 1``        ``if` `(n == 1)``            ``return` `F[0, 0] + F[0, 1];` `        ``power(F, n / 2);` `        ``multiply(F, F);` `        ``if` `(n % 2 != 0)``            ``multiply(F, M);` `        ``// Multiply it with initial values i.e``        ``// with F(0) = 0, F(1) = 1, F(2) = 1``        ``return` `F[0, 0] + F[0, 1];``    ``}` `    ``// Return n'th term of a series defined``    ``// using below recurrence relation.``    ``// f(n) is defined as``    ``// f(n) = f(n-1) + f(n-2) + f(n-3), n>=3``    ``// Base Cases :``    ``// f(0) = 0, f(1) = 1, f(2) = 1``    ``static` `int` `findNthTerm(``int` `n)``    ``{``        ``int``[, ] F = { { 1, 1, 1 }, { 1, 0, 0 },``                                 ``{ 0, 1, 0 } };` `        ``return` `power(F, n - 2);``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{``        ``int` `n = 5;``        ` `        ``Console.WriteLine(``"F(5) is "``                      ``+ findNthTerm(n));``    ``}``}` `// This code is contributed by vt_m.`

## PHP

 `= 3``// Base Cases :``// F(0) = 0, F(1) = 1, F(2) = 1` `// A utility function to multiply two matrices``// a[][] and b[][]. Multiplication result is``// stored back in b[][]``function` `multiply(&``\$a``, &``\$b``)``{``    ``// Creating an auxiliary matrix to store``    ``// elements of the multiplication matrix``    ``\$mul` `= ``array_fill``(0, 3,``           ``array_fill``(0, 3, 0));``    ``for` `(``\$i` `= 0; ``\$i` `< 3; ``\$i``++)``    ``{``        ``for` `(``\$j` `= 0; ``\$j` `< 3; ``\$j``++)``        ``{``            ``\$mul``[``\$i``][``\$j``] = 0;``            ``for` `(``\$k` `= 0; ``\$k` `< 3; ``\$k``++)``                ``\$mul``[``\$i``][``\$j``] += ``\$a``[``\$i``][``\$k``] *``                                ``\$b``[``\$k``][``\$j``];``        ``}``    ``}` `    ``// storing the multiplication result in a[][]``    ``for` `(``\$i` `= 0; ``\$i` `< 3; ``\$i``++)``        ``for` `(``\$j` `= 0; ``\$j` `< 3; ``\$j``++)``            ``\$a``[``\$i``][``\$j``] = ``\$mul``[``\$i``][``\$j``]; ``// Updating our matrix``}` `// Function to compute F raise to power n-2.``function` `power(``\$F``, ``\$n``)``{``    ``\$M` `= ``array``(``array``(1, 1, 1),``               ``array``(1, 0, 0),``               ``array``(0, 1, 0));` `    ``// Multiply it with initial values i.e with``    ``// F(0) = 0, F(1) = 1, F(2) = 1``    ``if` `(``\$n` `== 1)``        ``return` `\$F`` + ``\$F``;` `    ``power(``\$F``, (int)(``\$n` `/ 2));` `    ``multiply(``\$F``, ``\$F``);` `    ``if` `(``\$n` `% 2 != 0)``        ``multiply(``\$F``, ``\$M``);` `    ``// Multiply it with initial values i.e with``    ``// F(0) = 0, F(1) = 1, F(2) = 1``    ``return` `\$F`` + ``\$F`` ;``}` `// Return n'th term of a series defined``// using below recurrence relation.``// f(n) is defined as``// f(n) = f(n-1) + f(n-2) + f(n-3), n>=3``// Base Cases :``// f(0) = 0, f(1) = 1, f(2) = 1``function` `findNthTerm(``\$n``)``{``    ``\$F` `= ``array``(``array``(1, 1, 1),``               ``array``(1, 0, 0),``               ``array``(0, 1, 0));` `    ``return` `power(``\$F``, ``\$n` `- 2);``}` `// Driver code``\$n` `= 5;` `echo` `"F(5) is "` `. findNthTerm(``\$n``);` `// This code is contributed by mits``?>`

## Javascript

 ``

Output

`F(5) is 7`

Time Complexity: O(logN)
Auxiliary Space: O(logN)

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