Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Let $S$ be the set of all $3 \times 3$ symmetric matrices whose entries are either $0$ or $1$. Two of these entries are $1$ and four of them are $0$. A matrix is selected from set $S$, what is the probability that the selected matrix is non singular
Step-by-Step Solution
Key Concept: Count nonsingular $3 \times 3$ symmetric matrices with all diagonal elements equal to 1 and compute the probability by dividing nonsingular cases by total cases.
A $3 \times 3$ symmetric matrix with all diagonal elements equal to 1 has the form $\begin{pmatrix} 1 & a & b \\ a & 1 & c \\ b & c & 1 \end{pmatrix}$. For the matrix to be nonsingular, there are 12 such symmetric matrices total. Of these, 6 are nonsingular (those with determinant non-zero). Therefore, the probability that a randomly selected such matrix is nonsingular is $\frac{6}{12} = \frac{1}{2}$.
Correct Answer: 3