If | x x y x y x y x x |=0 then (a) y=x or y=-2 x (b) y=

step 1 So, in the given question we have the, there is a determinant of a matrix that is given as x x x

If | x x y x y x y x ...

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Key Concepts

Polynomial Factorization in Parameterized Determinants

When the determinant of a matrix is expressed as a polynomial in one or more parameters, finding the parameter values that make the determinant zero involves expanding the determinant and applying factorization techniques. Factorization allows the polynomial to be rewritten as a product of simpler expressions, whose roots indicate the specific conditions that lead to a singular matrix.

Determinant of a Matrix

A determinant is a scalar value computed from a square matrix that encodes important properties of the matrix, such as invertibility. A nonzero determinant indicates that the matrix is invertible, whereas a zero determinant indicates that the matrix is singular, meaning that its rows or columns are linearly dependent.

Matrix Singularity and Linear Dependence

A matrix is considered singular when its determinant equals zero. This implies that there is linear dependence among the rows or columns of the matrix, meaning one of them can be expressed as a combination of the others. This concept is crucial for understanding conditions where a system of equations may have either no solution or infinitely many solutions.

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