If a ≠0, a ≠1 and | x+1 x x x x+a x x x x+a^2 |=a^3+f(x)

step 1 In problem 60 we have this matrix that determine this matrix equals a cubed plus f of x multipli

If a ≠0, a ≠1 and | x+1 x x x x+a ...

If $a \neq 0, a \neq 1$ and
$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+a & x \\ x & x & x+a^{2}\end{array}\right|=a^{3}+f(x) \cdot a\left(a^{2}+a+1\right)$, then
(A) $f(x)=x$
(B) $f(x)=x^{2}$
(C) $f(x)=x^{3}$
(D) None of these

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Key Concepts

Matrix Determinant

The determinant is a scalar attribute of a square matrix that gives important information about the matrix, such as whether it is invertible. In the context of a 3x3 matrix, its determinant is calculated using a formula involving the sum and difference of products of its entries. This concept is key when transforming or simplifying matrix expressions, as well as in solving systems of linear equations and deriving polynomial expressions from matrix determinants.

Polynomial Identity and Coefficient Comparison

Polynomial identity involves setting two polynomial expressions equal to each other for all values of the variables involved. When a determinant expression results in a polynomial form, equating it to another polynomial allows one to compare the coefficients of like powers to solve for unknown functions or constants. This method, called coefficient comparison, is essential for determining identities and functions that satisfy given algebraic conditions.

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