Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
Let A be the set of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ c & a \end{pmatrix}$, such that $a,b,c \in \{0,1,2,3,4\}$ then:
Number of matrices in A which are symmetric or skew symmetric is 25
Number of matrices in A which are symmetric or skew symmetric & determinant divisible by 5 is 9
Number of matrices in A for which trace is not divisible by 5 & determinant divisible by 5 is 16
Number of matrices in A for which determinant is not divisible by 5 is 109
Step-by-Step Solution
Key Concept: Symmetric matrices have determinant divisible by 5 when $(a+b)(a-b)$ is divisible by 5, which occurs for specific value pairs.
For symmetric matrices $x = \begin{pmatrix} a & b \\ b & a \end{pmatrix}$, there are $5^2 = 25$ matrices. The determinant is $|x| = a^2 - b^2 = (a+b)(a-b)$. For divisibility by 5, either $a - b = 0$ (i.e., $a = b$) or $a + b = 5$ with $a, b ∈ \{0, 1, 2, 3, 4\}$. This gives 5 matrices where $a = b$, plus 4 matrices each from $a + b = 5$. For skew-symmetric matrices, only the zero matrix is possible. Trace $2a$ is never divisible by 5 when $a ≠ 0$.
Correct Answer: 1,2,3