The value of the determinant | √(13)+√(3) 2 √(5) √(5) √(15)+√(26) 5 10

step 1 In problem 33 we want to get the value of the determinant square out of 13 plus square to 3 2 sq

The value of the determinant | √(13)+√(3) 2 √(5) ...

The value of the determinant $\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & 10 \\ 3+\sqrt{65} & \sqrt{15} & 5\end{array}\right|$ is equal to: (A) $5 \sqrt{3}(\sqrt{6}-5)$ (B) $5 \sqrt{3}(\sqrt{6}-\sqrt{5})$ (C) $5(\sqrt{6}-5)$ (D) $\sqrt{3}(\sqrt{6}-\sqrt{5})$

The value of the determinant $\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & 10 \\ 3+\sqrt{65} & \sqrt{15} & 5\end{array}\right|$
is equal to:
(A) $5 \sqrt{3}(\sqrt{6}-5)$
(B) $5 \sqrt{3}(\sqrt{6}-\sqrt{5})$
(C) $5(\sqrt{6}-5)$
(D) $\sqrt{3}(\sqrt{6}-\sqrt{5})$

Key Concepts

Algebraic Simplification

Algebraic simplification is the process of reducing complex expressions, including those with radicals or irrational numbers, to a simpler form. This is particularly important in determinant evaluations that involve complicated numeric expressions, ensuring that the final result is clear and manageable.

Determinant

The determinant is a scalar attribute associated with a square matrix that provides insight into properties such as invertibility and volume scaling in linear transformations. It is computed through specific arithmetic combinations of the matrix elements and is essential to solving systems of linear equations.

Cofactor Expansion

Cofactor expansion (or Laplace expansion) is a method for evaluating the determinant of a matrix by breaking it down into smaller determinants. This technique simplifies the computation by expanding along a row or column, reducing a 3x3 determinant into combinations of 2x2 determinants.

Matrix Row Operations

Matrix row operations involve systematically manipulating the rows of a matrix to simplify calculations. Operations such as row swapping, multiplying a row by a constant, and adding multiples of one row to another can transform the matrix into a form where calculating the determinant becomes more straightforward, while keeping track of how these operations affect the determinant's value.

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