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In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then it can be decomposed into n determinants, where n has value
- a)1
- b)9
- c)24
- d)16
Correct answer is option 'C'. Can you explain this answer?
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In a third order determinant, each element of the first column consist...
N = 2 ×3 × 4 = 24.
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In a third order determinant, each element of the first column consist...
**Given information:**
- Each element of the first column consists of the sum of two terms.
- Each element of the second column consists of the sum of three terms.
- Each element of the third column consists of the sum of four terms.
**To find:**
The number of determinants in which the given third-order determinant can be decomposed.
**Solution:**
Let's denote the given third-order determinant as D.
D = |a1 b1 c1|
|a2 b2 c2|
|a3 b3 c3|
Since each element of the first column consists of the sum of two terms, we can write:
a1 = p + q,
a2 = r + s,
a3 = t + u,
Similarly, for the second and third columns, we have:
b1 = v + w + x,
b2 = y + z + p,
b3 = q + r + s,
c1 = t + u + v + w,
c2 = x + y + z + p,
c3 = q + r + s + t.
Now, let's expand the determinant D using the first row:
D = a1(b2c3 - b3c2) - b1(a2c3 - a3c2) + c1(a2b3 - a3b2)
Expanding each term, we get:
D = (p + q)(y + z + p)(q + r + s + t) - (v + w + x)(r + s)(q + r + s + t) + (t + u + v + w)(r + s)(q + r + s + t)
Now, let's simplify this expression:
D = (p + q)(y + z + p)(q + r + s + t) - (v + w + x)(r + s)(q + r + s + t) + (t + u + v + w)(r + s)(q + r + s + t)
Distributing the terms, we get:
D = pq(y + z + p)(q + r + s + t) + p(y + z + p)(r + s)(q + r + s + t) + (v + w + x)(r + s)(q + r + s + t)(q + r + s + t) + (t + u + v + w)(r + s)(q + r + s + t)
Now, we can see that D can be decomposed into several determinants, each containing only one term.
The total number of terms in the expanded expression is 2 * 3 * 4 = 24, which means that D can be decomposed into 24 determinants.
Therefore, the correct answer is option **C) 24**.
- Each element of the first column consists of the sum of two terms.
- Each element of the second column consists of the sum of three terms.
- Each element of the third column consists of the sum of four terms.
**To find:**
The number of determinants in which the given third-order determinant can be decomposed.
**Solution:**
Let's denote the given third-order determinant as D.
D = |a1 b1 c1|
|a2 b2 c2|
|a3 b3 c3|
Since each element of the first column consists of the sum of two terms, we can write:
a1 = p + q,
a2 = r + s,
a3 = t + u,
Similarly, for the second and third columns, we have:
b1 = v + w + x,
b2 = y + z + p,
b3 = q + r + s,
c1 = t + u + v + w,
c2 = x + y + z + p,
c3 = q + r + s + t.
Now, let's expand the determinant D using the first row:
D = a1(b2c3 - b3c2) - b1(a2c3 - a3c2) + c1(a2b3 - a3b2)
Expanding each term, we get:
D = (p + q)(y + z + p)(q + r + s + t) - (v + w + x)(r + s)(q + r + s + t) + (t + u + v + w)(r + s)(q + r + s + t)
Now, let's simplify this expression:
D = (p + q)(y + z + p)(q + r + s + t) - (v + w + x)(r + s)(q + r + s + t) + (t + u + v + w)(r + s)(q + r + s + t)
Distributing the terms, we get:
D = pq(y + z + p)(q + r + s + t) + p(y + z + p)(r + s)(q + r + s + t) + (v + w + x)(r + s)(q + r + s + t)(q + r + s + t) + (t + u + v + w)(r + s)(q + r + s + t)
Now, we can see that D can be decomposed into several determinants, each containing only one term.
The total number of terms in the expanded expression is 2 * 3 * 4 = 24, which means that D can be decomposed into 24 determinants.
Therefore, the correct answer is option **C) 24**.
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