If | 1+x x x^2 x 1+x x^2 x^2 x 1+x |=p x^5+q x^4+r x^3+s x^

step 1 I have given here left -in -side is this a determinant equal to this given polynomial. We're goi

If | 1+x x x^2 x 1+x x^2 x^2 ...

If $\left|\begin{array}{ccc}1+x & x & x^{2} \\ x & 1+x & x^{2} \\ x^{2} & x & 1+x\end{array}\right|=p x^{5}+q x^{4}+r x^{3}+s x^{2}+t x+w$, then $$ \begin{array}{ll} \hline \text { Column-I } & \text { Column-II } \\ \hline \text { I. } w \text { is equal to } & \text { (A) } 3 \\ \text { II. } t \text { is equal to } & \text { (B) } 1 \\ \text { III. } p+r \text { is equal to } & \text { (C) }-1 \\ \text { IV. } q+s \text { is equal to } & \text { (D) } 0 \\ \hline \end{array} $$

If $\left|\begin{array}{ccc}1+x & x & x^{2} \\ x & 1+x & x^{2} \\ x^{2} & x & 1+x\end{array}\right|=p x^{5}+q x^{4}+r x^{3}+s x^{2}+t x+w$, then
$$
\begin{array}{ll}
\hline \text { Column-I } & \text { Column-II } \\
\hline \text { I. } w \text { is equal to } & \text { (A) } 3 \\
\text { II. } t \text { is equal to } & \text { (B) } 1 \\
\text { III. } p+r \text { is equal to } & \text { (C) }-1 \\
\text { IV. } q+s \text { is equal to } & \text { (D) } 0 \\
\hline
\end{array}
$$

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Key Concepts

Determinant Calculation

Determinant calculation involves finding a scalar value from a square matrix that encapsulates important properties of the matrix, such as whether it is singular or invertible. This often involves methods like cofactor expansion or other systematic approaches that reduce the matrix size recursively to reveal the complete determinant.

Parametric Matrices

Parametric matrices are matrices in which the entries depend on one or more variables or parameters. This dependency means that the determinant itself becomes a function of these parameters, allowing the study of how changes in the parameters influence the overall behavior of the matrix, such as its invertibility or the nature of its eigenvalues.

Polynomial Coefficient Identification

After expanding the determinant of a parametric matrix, the resulting expression is typically a polynomial. Identifying the coefficients of terms in this polynomial by equating them to corresponding terms in a given polynomial form allows one to determine specific numerical relationships and values. This process of coefficient comparison is fundamental in matching the expansion to a standardized polynomial form.

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