If Δ=| 1 2 3 2 -1 0 3 4 5 | and Δ^'=| 1 6 3

step 1 So in the given question we are given two determinants. The first one is delta which is given as

If Δ=| 1 2 3 2 -1 0 3 4 ...

If $\Delta=\left|\begin{array}{ccc}1 & 2 & 3 \\ 2 & -1 & 0 \\ 3 & 4 & 5\end{array}\right|$ and $\Delta^{\prime}=\left|\begin{array}{ccc}1 & 6 & 3 \\ 4 & -6 & 0 \\ 3 & 12 & 5\end{array}\right|$, the $\Delta^{\prime} \div \Delta$ is equal to
(a) 2
(b) 3
(c) 6
(d) 0

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Key Concepts

Properties of Determinants Under Elementary Operations

Determinants have several key properties that relate to elementary row and column operations. For instance, multiplying a row or column by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another row does not change the determinant. These properties are essential when simplifying determinant calculations or understanding how modifications to a matrix affect its determinant.

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that provides important information about the matrix, such as whether it is invertible and how it scales volumes in linear transformations. It is a fundamental concept in linear algebra and is used to solve systems of linear equations, compute eigenvalues, and analyze matrix behavior in various contexts.

Expansion by Cofactors

Expansion by cofactors is a method used to calculate the determinant of a matrix by breaking it down along a row or column into smaller parts. Each term in the expansion is the product of an element of the matrix and its corresponding cofactor, which involves a minor determinant and a sign factor. This method is particularly useful for manually computing determinants of matrices of sizes 3×3 and larger.

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