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प्रश्न
if `|(a, b, c),(m, n, p),(x, y, z)| = k`, then what is the value of `|(6a, 2b, 2c),(3m, n, p),(3x, y, z)|`?
विकल्प
`k/6`
2k
3k
6k
उत्तर
6k
Explanation:
Given:
`|(a, b, c),(m, n, p),(x, y, z)| = k`
We want to find:
`|(6a, 2b, 2c),(3m, n, p),(3x, y, z)|`
Applying the Scaling Rule
- The first row is scaled by 6, 2 and 2 for a, b and c respectively. This results in a scaling factor of 6 × 2 × 2 = 24.
- The second row is scaled by 3 for m. This results in an additional scaling factor of 3.
- The third row is scaled by 3 for x. This results in an additional scaling factor of 3.
So, the total scaling factor is:
24 × 3 × 3 = 216
However, we need to scale the entire first row by 2 to maintain the proportions:
`|(6a, 2b, 2c),(3m, n, p),(3x, y, z)| = 6 * |(6a, 2/6b, 2/6c),(m, 1/3n, 1/3p),(3x, 1/3y, 1/3z)|`
Therefore, the value of the determinant is:
= `6|(a, b, c),(m, n, p),(x, y, z)|`
= 6k
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