Let α and β be the distinct roots of the equation x² + x - 1 = 0. Consider the set T = {1, α, β}. Fo

Q: Wait, why is the sum of elements in $T$ equal to zero?

Great observation! The roots of $x^{2} + x - 1 = 0$ are $\alpha$ and $\beta$. By Vieta's formulas, the sum of the roots is $\alpha + \beta = -\frac{1}{1} = -1$. Therefore, $1 + \alpha + \beta = 0$. Since $T = \{1, \alpha, \beta\}$, the sum of all elements in $T$ is indeed zero.

Q: Why are there exactly 12 matrices in Part P?

Think of it as a permutation problem. The first row can be any permutation of $\{1, \alpha, \beta\}$, which gives $3! = 6$ ways. For the second row, to ensure column sums are zero, it must be a derangement of the first row. There are exactly 2 such derangements. Once the first two rows are fixed, the third row is uniquely determined to satisfy the column sum condition. Thus, $6 \times 2 \times 1 = 12$.

Q: Why is the determinant of the skew-symmetric matrix zero?

A fundamental property of skew-symmetric matrices $M$ is that $M^{T} = -M$. For a $3 \times 3$ matrix, $\det(M) = \det(M^{T}) = \det(-M) = (-1)^{3} \det(M) = -\det(M)$. This implies $2 \det(M) = 0$, so $\det(M) = 0$. This is a powerful tool in linear algebra!

Question

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^{2} + x - 1 = 0$. Consider the set $T = \{1, \alpha, \beta\}$. For a $3 	imes 3$ matrix $M = (a_{ij})_{3	imes3}$, define $R_{i} = a_{i1} + a_{i2} + a_{i3}$ and $C_{j} = a_{1j} + a_{2j} + a_{3j}$ for $i = 1, 2, 3$ and $j = 1, 2, 3$. Match each entry in List-I to the correct entries in List-II.

List-I:

P. The number of matrices $M = (a_{ij})_{3	imes3}$ with all entries in $T$ such that $R_{i} = C_{j} = 0$ for all $i, j$, is

Q. The number of symmetric matrices $M = (a_{ij})_{3	imes3}$ with all entries in $T$ such that $C_{j} = 0$ for all $j$, is

R. Let $M = (a_{ij})_{3	imes3}$ be a skew symmetric matrix such that $a_{ij} \in T$ for $i > j$. Then the number of elements in the set $\left\{ \begin{pmatrix} x \ y \ z \end{pmatrix} : x, y, z \in \mathbb{R}, M \begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} a_{12} \ 0 \ -a_{23} \end{pmatrix} ight\}$ is

S. Let $M = (a_{ij})_{3	imes3}$ be a matrix with all entries in $T$ such that $R_{i} = 0$ for all $i$. Then the absolute value of determinant of $M$ is

List-II:

1. 1

2. 12

3. infinite

4. 6

5. 0

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We begin with the quadratic equation $x^{2} + x - 1 = 0$. Its roots, $\alpha$ and $\beta$, serve as the building blocks of the set $T = \{1, \alpha, \beta\}$.

From Vieta's formulas, we know that $\alpha + \beta = -1$. This leads us to the master identity:
$$1 + \alpha + \beta = 0$$

This identity is the heartbeat of the problem, confirming that the sum of the elements in set $T$ is zero. This property will simplify all subsequent matrix operations.

### The Matrix Puzzle: Part P

We construct a $3 	imes 3$ matrix $M$ where every row sum $R_i$ and every column sum $C_j$ equals zero. Because the sum of elements in $T$ is zero, each row and column must contain exactly one instance of $1$, $\alpha$, and $\beta$.

The first row can be any permutation of these three elements, providing $3! = 6$ possibilities. For the second row, we must ensure no element is repeated in any column, which requires the second row to be a derangement of the first.

For three elements, there are exactly two such derangements. Once the first two rows are fixed, the third row is uniquely determined to satisfy the zero-sum condition. Thus, the total number of such matrices is:
$$6 	imes 2 	imes 1 = 12$$

### Symmetry and Constraints: Part Q

Next, we consider symmetric matrices where $C_j = 0$. A symmetric matrix satisfies $M = M^{T}$, which implies $R_i = C_i$.

If the column sums are zero, the row sums are automatically zero. We seek the subset of our 12 matrices that are symmetric. For a matrix to be symmetric, the elements across the main diagonal must be equal.

If we fix the first row, there is only one way to arrange the remaining elements to maintain symmetry while satisfying the zero-sum condition. Since there are 6 choices for the first row, we find exactly **6 symmetric matrices**.

### The Skew-Symmetric Mystery: Part R

We now examine skew-symmetric matrices, where the diagonal elements are zero and $a_{ij} = -a_{ji}$. We consider the system of equations $M \mathbf{x} = \mathbf{b}$.

Because $M$ is a $3 	imes 3$ skew-symmetric matrix, its determinant is zero. This implies that the system is linearly dependent.

When we expand the matrix multiplication, we obtain three linear equations. These equations are consistent, and because the determinant is zero, the system does not have a unique solution. Instead, it possesses **infinite solutions**.

### The Determinant's Vanishing Act: Part S

Finally, we calculate the determinant of a matrix where $R_i = 0$. We apply the column operation $C_1 	o C_1 + C_2 + C_3$.

Because the sum of elements in each row is zero, the new first column becomes a column of zeros. A matrix with a column of zeros has a determinant of zero.

Thus, the absolute value of the determinant is **0**. This concludes our journey through the properties of roots, permutations, and matrix algebra.

List-I

List-II

Select Matching Pairs:

Visualized Solution

Roots and Set $T$

Part P: $R_{i} = C_{j} = 0$

Counting Matrices for Part P

Part Q: Symmetric Matrices

Counting Symmetric Matrices

Part R: Skew-Symmetric Matrix

Expanding the Linear System

Infinite Solutions

Part S: Determinant with $R_{i} = 0$

Column Operation and Determinant

Final Matches

The Sigma Insight: Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

Analyzing the Setup

The Matrix Puzzle

Symmetry and Constraints

The Skew-Symmetric Mystery

The Determinant's Vanishing Act

Similar Questions

Comprehension Passage

The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

Let $A$ be a $3 \times 3$ real matrix such that $A 101 = 2101, A - 1 01 = 4 - 1 01, A 010 = 2010$ . Then, the system $(A - 3 I) x y z = 123$ has

The number of $3 \times 3$ matrices $A$ whose entries are either 0 or 1 and for which the system $A x y z = 100$ has exactly two distinct solutions, is

Let $M = (a_{ij}), i, j \in {1, 2, 3}$ , be the $3 \times 3$ matrix such that $a_{ij} = 1$ if $j + 1$ is divisible by $i$ , otherwise $a_{ij} = 0$ . Then which of the following statements is (are) true ?

Comprehension Passage

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is symmetric matrix and $B$ is skew-symmetric matrix. Then the system of linear equations $(A^{2} B^{2} - B^{2} A^{2}) X = O$ , where $X$ is a $3 \times 1$ column matrix of unknown variables and $O$ is a $3 \times 1$ null matrix, has :

Let $A = 211010101$ , $B = [B_{1}, B_{2}, B_{3}]$ , where $B_{1}, B_{2}, B_{3}$ are column matrices, and $A B_{1} = 100$ , $A B_{2} = 230$ , $A B_{3} = 321$ . If $α = ∣ B ∣$ and $β$ is the sum of all the diagonal elements of $B$ , then $α^{3} + β^{3}$ is equal to

For $α, β \in R$ , suppose the system of linear equations $x - y + z = 5$ , $2 x + 2 y + α z = 8$ , $3 x - y + 4 z = β$ has infinitely many solutions. Then $α$ and $β$ are the roots of

Let $S_{1}$ and $S_{2}$ be respectively the sets of all $a \in R - {0}$ for which the system of linear equations $a x + 2 a y - 3 a z = 1$ , $(2 a + 1) x + (2 a + 3) y + (a + 1) z = 2$ , $(3 a + 5) x + (a + 5) y + (a + 2) z = 3$ has unique solution and infinitely many solutions. Then

List-I

List-II

Select Matching Pairs:

Visualized Solution

Roots and Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T

Part P: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Ri​=Cj​=0

Counting Matrices for Part P

Part Q: Symmetric Matrices

Counting Symmetric Matrices

Part R: Skew-Symmetric Matrix

Expanding the Linear System

Infinite Solutions

Part S: Determinant with .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Ri​=0

Column Operation and Determinant

Final Matches

The Sigma Insight: Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

Analyzing the Setup

The Matrix Puzzle

Symmetry and Constraints

The Skew-Symmetric Mystery

The Determinant's Vanishing Act

Similar Questions

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices in A is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ has a unique solution, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ is inconsistent, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of 3×3 matrices A whose entries are either 0 or 1 and for which the system A​xyz​​=​100​​ has exactly two distinct solutions, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let M=(aij​),i,j∈{1,2,3}, be the 3×3 matrix such that aij​=1 if j+1 is divisible by i, otherwise aij​=0. Then which of the following statements is (are) true ?

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a solution of x3−1=0 with Im(ω)>0, if a=2 with b and c satisfying (E), then the value of ωa3​+ωb1​+ωc3​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let b=6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax2+bx+c=0, then ∑n=0∞​(α1​+β1​)n is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For α,β∈R, suppose the system of linear equations x−y+z=5, 2x+2y+αz=8, 3x−y+4z=β has infinitely many solutions. Then α and β are the roots of

Roots and Set $T$

Part P: $R_{i} = C_{j} = 0$

Part S: Determinant with $R_{i} = 0$

The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

The number of $3 \times 3$ matrices $A$ whose entries are either 0 or 1 and for which the system $A x y z = 100$ has exactly two distinct solutions, is

Let $M = (a_{ij}), i, j \in {1, 2, 3}$ , be the $3 \times 3$ matrix such that $a_{ij} = 1$ if $j + 1$ is divisible by $i$ , otherwise $a_{ij} = 0$ . Then which of the following statements is (are) true ?

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

For $α, β \in R$ , suppose the system of linear equations $x - y + z = 5$ , $2 x + 2 y + α z = 8$ , $3 x - y + 4 z = β$ has infinitely many solutions. Then $α$ and $β$ are the roots of