Question

Consider `Δ = |(2a, 2b, 2c),(2e, f, g),(2i, j, k)|`

Assertion: The value of `Δ = 4 xx |(a, b, c),(e, f, g),(i, j, k)|`

Reason: If all elements of one row or one column of a determinant are multiplied by a scalar, k then the value of the determinant is multiplied by k.

Which of the following is correct?

पर्याय

• Both Assertion and Reason are true and Reason is the correct explanation for Assertion.

• Both Assertion and Reason are true, but Reason is not the correct explanation for Assertion.

• Assertion is true and Reason is false.

• Assertion is false and Reason is true.

Answer 1

Assertion is false and Reason is true.

Explanation:

Reason: The reason correctly states a fundamental property of determinants. If all elements of one row or one column are multiplied by a scalar k, the value of the determinant is multiplied by k.

This is a true statement.

Assertion: The assertion claims that `Δ = 4 xx |(a, b, c),(e, f, g),(i, j, k)|`

The original determinant is `Δ = |(2a, 2b, 2c),(2e, f, g),(2i, j, k)|`

We can factor out a 2 from the first row:

`Δ = 2 xx |(a, b, c),(2e, f, g),(2i, j, k)|`

Only the first row has all its elements multiplied by 2, not all the rows in the entire determinant.

The elements f, g, j, k in the second and third rows are not multiplied by 2 in the original determinant Δ.

Therefore, the total value is only multiplied by 2, not 4.

The assertion is false.