# Dot Product of Diagonals

This challenge is very simple. You are given as input a square matrix, represented in any sane way, and you have to output the dot product of the diagonals of the matrix.

The diagonals in specific are the diagonal running from top-left to bottom-right and from top-right to bottom-left.

# Test Cases

[[-1, 1], [-2, 1]]  ->  -3
[[824, -65], [-814, -741]]  ->  549614
[[-1, -8, 4], [4, 0, -5], [-3, 5, 2]]  ->  -10
[[0, -1, 0], [1, 0, 2], [1, 0, 1]]  ->  1


# Jelly, 8 6 bytes

×UŒDḢS


### How it works

×UŒDḢS  Main link. Argument: M (matrix)

U      Upend M, i.e., reverse each row.
,       Pair M and upended M.
ŒD    Yield all diagonals.
Ḣ   Head; extract the first, main diagonal.
S  Reduce by sum.


# MATL, 8 bytes

t!P!*Xds


Input format is

[-1, -8, 4; 4, 0 -5; -3, 5, 2]


### Explanation

t       % Take input matrix implicitly. Duplicate
!P!     % Flip matrix horizontally
*       % Element-wise product
Xd      % Extract main diagonal as a column vector
s       % Sum. Display implicitly


# Python, 47 bytes

lambda x:sum(r[i]*r[~i]for i,r in enumerate(x))


Test it on Ideone.

# J, 21 19 bytes

[:+/(<0 1)|:(*|."1)


Straight-forward approach.

Saved 2 bytes thanks to @Lynn.

## Usage

The input array is shaped using dimensions $values.  f =: [:+/(<0 1)|:(*|."1) f (2 2$ _1 1 _2 1)
_3
f (2 2 $824 _65 _814 _741) 549614 f (3 3$ _1 _8 4 4 0 _5 _3 5 2)
_10
f (3 3 \$ 0 _1 0 1 0 2 1 0 1)
1


## Explanation

[:+/(<0 1)|:(*|."1)    Input: matrix M
|."1     Reverse each row of M
*         Multiply element-wise M and the row-reversed M
(<0 1)|:           Take the diagonal of that matrix
[:+/                   Sum that diagonal and return it=

• [:+/(<0 1)|:(*|."1) is 19 bytes – Lynn Jun 11 '16 at 14:24

# Julia, 25 bytes

~=diag
!x=~x⋅~rotl90(x)


Try it online!

• rot90, good idea! – Luis Mendo Jun 11 '16 at 0:26

## JavaScript (ES6), 45 bytes

a=>a.reduce((r,b,i)=>r+b[i]*b.slice(~i)[0],0)
a=>a.reduce((r,b,i)=>r+b[i]*b[b.length+~i],0)


## R, 26 bytes

sum(diag(A*A[,ncol(A):1]))


# Mathematica, 17 bytes

Tr[#~Reverse~2#]&


# APL (Dyalog), 15 9 bytes

+/1 1⍉⌽×⊢


Try it online!

How?

+/ - sum

1 1⍉ - diagonal of

⌽×⊢ - element wise multiplication of the matrix with it's reverse

## Clojure, 57 bytes

#(apply +(map(fn[i r](*(r i)(nth(reverse r)i)))(range)%))


I liked my previous solution more, but this is much shorter (basically does the same as the Python solution):

f m=sum[r!!i*r!!(length m-i-1)|(i,r)<-zip[0..]m]


Try it online!

# J, 18 Bytes

<:@#{+//.@:(*|."1)


### Explaination:

           (     ) | Monadic hook
*      | Argument times...
|."1  | The argument mirrored around the y axis
+//.@:        | Make a list by summing each of the diagonals of the matrix
{              | Takes element number...
<:@#               | Calculates the correct index (size of the array - 1)


# 05AB1E, 5 bytes

í*Å\O


Explanation:

í        # Reverse each row of the (implicit) input-matrix
#  i.e. [[-1,-8,4],[4,0,-5],[-3,5,2]] → [[4,-8,-1],[-5,0,4],[2,5,-3]]
*       # Multiply it with the (implicit) input-matrix (at the same positions)
#  i.e. [[-1,-8,4],[4,0,-5],[-3,5,2]] and [[4,-8,-1],[-5,0,4],[2,5,-3]]
#   → [[-4,64,-4],[-20,0,-20],[-6,25,-6]]
Å\     # Get the diagonal-list from the top-left corner towards the bottom-right
#  i.e. [[-4,64,-4],[-20,0,-20],[-6,25,-6]] → [-4,0,-6]
O    # Sum it (and output implicitly)
#  i.e. [-4,0,-6] → -10