Question

Let \(S=\left\{\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}: a_{ij}\in\{0,1,2\},\ a_{11}=a_{22}\right\}\). Then the number of non-singular matrices in the set \(S\) is :

• A. 27
• B. 24
• C. 10
• D. 20

Hint:

Count the singular ones: \(a^2=bc\) with all entries from \(\{0,1,2\}\).

Answer 1

The Correct Option is D

Let the matrix be \(\begin{pmatrix}a&b\\c&a\end{pmatrix}\) with \(a,b,c\in\{0,1,2\}\): total \(27\) matrices. Determinant \(=a^2-bc\).
Singular when \(a^2=bc\).
\(a=0\): \(bc=0\): \(5\) pairs (b=0: 3 choices of c; c=0 with b=1,2: 2 more).
\(a=1\): \(bc=1\): \((1,1)\) only, 1 pair.
\(a=2\): \(bc=4\): \((2,2)\) only, 1 pair.
Singular \(=5+1+1=7\); non-singular \(=27-7=20\).
Answer: (4) 20