MATLAB: Inv(matrix) takes shorter time than \ operator

MATLAB

A=magic(5)
A =
      17    24     1     8    15
      23     5     7    14    16
       4     6    13    20    22
      10    12    19    21     3
      11    18    25     2     9
>> tic
inv(A)
toc
ans =
     -0.0049    0.0512   -0.0354    0.0012    0.0034
      0.0431   -0.0373   -0.0046    0.0127    0.0015
     -0.0303    0.0031    0.0031    0.0031    0.0364
      0.0047   -0.0065    0.0108    0.0435   -0.0370
      0.0028    0.0050    0.0415   -0.0450    0.0111
Elapsed time is 0.006097 seconds.
>> tic
B=eye(5);
A\B
toc
ans =
     -0.0049    0.0512   -0.0354    0.0012    0.0034
      0.0431   -0.0373   -0.0046    0.0127    0.0015
     -0.0303    0.0031    0.0031    0.0031    0.0364
      0.0047   -0.0065    0.0108    0.0435   -0.0370
      0.0028    0.0050    0.0415   -0.0450    0.0111
Elapsed time is 0.008993 seconds.
Just a general question, shouldn't have been the other way round? inv(matrix) is supposed to be more computationally intensive than the \ operation.

Best Answer

While inv indeed does involve more operations than \ because it involves inverse as well as multiplication, \ is just one operation. It is safer as you can use it on non-full rank of the system which is determined by QR decomp. I persuade you to look at this literature from Cleve the creator of MATLAB: Linear Equation and of course this link in the documentation.
To support what James said, your example is a very bad one
Not the best but a supporting example is as follows. Using a larger matrix, and making sure the random numbers are the same
rng(5);tic, inv(rand(1000))*rand(1000,1); toc
rng(5);tic, rand(1000)\rand(1000,1); toc
Elapsed time is 0.321691 seconds.
Elapsed time is 0.142591 seconds.
You will have to run this a couple of times. The first run may result in MATLAB performing some optimizations and memory allocations.
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