QJEEM23Maf45e: JEE Main 2023 Mathematics - Class 12 JEEM23Maf45e Solution: The number of symmetric matrices of order 3, with all the…

Sure! A symmetric matrix is a square matrix that is equal to its transpose. For a matrix to be symmetric, the element at row i and column j must be equal to the element at row j and column i. In other words, $A_{ij} = A_{ji}$ .

For a 3 $\times$ 3 symmetric matrix, it looks like this:

$\begin{pmatrix}a & b & c \\b & d & e \\c & e & f \\\end{pmatrix}$

Notice that there are only 6 unique elements we need to fill because of the symmetry:

1. $a$ in the (1,1) position
2. $b$ in the (1,2) and (2,1) positions
3. $c$ in the (1,3) and (3,1) positions
4. $d$ in the (2,2) position
5. $e$ in the (2,3) and (3,2) positions
6. $f$ in the (3,3) position

Each of these unique elements can take a value from the set ${0,1,2,3,4,5,6,7,8,9}$ , which has 10 elements.

We have 10 choices for each of the 6 unique elements, so the total number of symmetric matrices can be calculated as:

$10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^{6}$

Thus, the total number of symmetric matrices of order 3 with entries from this set is $10^{6}$ .