Inspired by Comrade Soumava Pal

Algebra Level 5

How many invertible real 3 × 3 3\times3 matrices A A are there whose entries are all 1 1 or − 1 -1 .

Clarification : A matrix A A is invertible if det ⁡ ( A ) ≠ 0 \det(A)\neq 0 .

Inspiration .


The answer is 192.

Otto Bretscher by Otto Bretscher , 65, USA

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2 solutions

Joe Mansley
Jul 25, 2018

The matrix will be invertible, the column vectors must be linearly independent. There are 4 linearly independent vectors you can make using only 1s and -1s. We must choose 3 of these to make a matrix. We have 4 choices for the 1st column, 3 choices for the 2nd and 2 choices for the 3rd. For each of the column vectors you have the option of multiplying it by -1, and it will still be independent of the others, so the answer is 4x3x2x2^3=192

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Rajdeep Brahma
Jun 11, 2018

Choices of 1st column of non-zero vector is 2 3 2^3 =8,Choices for 2nd column is (8-2)=6,Choices for 3rd column is (8-4)=4,Total number= 8 ∗ 6 ∗ 4 8*6*4 =192.

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