Animated Solution for Mathematics - Matrices and Determinants: Comprehension Passage

Unlocking the Matrix

Welcome, future engineer. Today, we are not just solving a matrix problem; we are exploring the architecture of symmetry.

When you look at a symmetric matrix, you are looking at a mirror. The elements across the main diagonal are not independent entities; they are reflections. This simple geometric constraint is the key that will unlock this entire problem.

The Combinatorial Puzzle

Imagine a $3\times 3$ grid. We have nine slots to fill, using five '1's and four '0's.

Because the matrix $A$ is symmetric, if we place a '1' at position $(i,j)$, we are forced to place a '1' at $(j,i)$. This means our off-diagonal elements must appear in pairs.

Let $k$ be the number of '1's on the main diagonal. The remaining $(5-k)$ ones must occupy the off-diagonal positions. Since these must form pairs, $(5-k)$ must be an even number.

Since $k$ is the number of diagonal elements and there are only three diagonal spots, $k$ must be either $1$ or $3$. This parity argument splits our problem into two manageable cases.

Counting the Possibilities

In Case 1, where $k=1$, we choose one diagonal spot in $\left(\genfrac{}{}{0ex}{}{3}{1}\right)=3$ ways. We have four '1's left, which form two pairs. We choose two pairs out of the three available off-diagonal pairs in $\left(\genfrac{}{}{0ex}{}{3}{2}\right)=3$ ways.

Thus, there are $3\times 3=9$ such matrices.

In Case 2, where $k=3$, all diagonal elements are '1'. We have two '1's left, forming one pair. We choose one pair out of three in $\left(\genfrac{}{}{0ex}{}{3}{1}\right)=3$ ways.

The total number of matrices is $9+3=12$. We have mapped the entire landscape of our set $\mathcal{A}$.

The Determinant's Secret

Now, we turn to the system:

A unique solution exists if and only if $|A| eq 0$. Let us write the determinant for a general symmetric matrix:

The expansion is $|A|=abc+2def-a{f}^2-b{e}^2-c{d}^2$.

Because our entries are only $0$ or $1$, we know that $x^2=x$. This allows us to simplify the determinant to:

This is a powerful tool. It turns a cubic expression into a simple sum and difference of binary variables.

Unique Solutions and Inconsistency

We test our 12 matrices. For Case 2 ($k=3$), we find that $|A|=0$ for all three matrices; they are all singular.

For Case 1 ($k=1$), we find that six matrices have $|A|=-1$ (non-zero) and three have $|A|=0$. Thus, exactly six matrices yield a unique solution.

Finally, we tackle inconsistency. A system is inconsistent if $|A|=0$ and the system has no solution.

We examine the six matrices where $|A|=0$. By writing out the equations for each, we find that in some cases, we get contradictions like $0=1$ (inconsistent), while in others, we get redundant equations like $0=0$ (infinite solutions).

We discover that two matrices from Case 2 and two from Case 1 are inconsistent, totaling four inconsistent matrices.

Mathematics is not about memorizing formulas; it is about observing constraints and simplifying the complex. You have just navigated a combinatorial and algebraic maze with precision. Keep this systematic approach in your toolkit, and no JEE problem will ever be too daunting.