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If each element in a row of a determinant is multiplied by the same factor r, then the value of the determinant
- a)is multiplied by r3
- b)is increased by 3r
- c)remains unchanged
- d)is multiplied by r
Correct answer is option 'D'. Can you explain this answer?
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If each element in a row of a determinant is multiplied by the same fa...
Multiplication of a Row by a Constant in a Determinant
When a row of a determinant is multiplied by a constant k, the determinant changes, but there is a simple relationship between the original determinant and the new one. Specifically, if the determinant is A and the new determinant is B, then B can be obtained from A by multiplying one row of A by k, as follows:
- If a row of A is multiplied by k, then the corresponding element of B is k times the original element.
- All other elements of B are the same as the corresponding elements of A.
- Thus, B is obtained by replacing one row of A by its multiple by k.
Effect of Multiplication of a Row by a Constant on the Value of a Determinant
The value of a determinant is a scalar quantity that depends on the elements of the determinant. When a row of a determinant is multiplied by a constant k, the value of the determinant changes by a factor of k. Specifically, if the determinant is A and the new determinant is B, then the value of B is k times the value of A.
Examples
Example 1:
Consider the following determinant:
| 1 2 3 |
| 4 5 6 |
| 7 8 9 |
If the second row is multiplied by 3, the determinant becomes:
| 1 2 3 |
| 12 15 18 |
| 7 8 9 |
The value of the original determinant is:
1(5×9−6×8)−2(4×9−6×7)+3(4×8−5×7)=0
The value of the new determinant is:
1(15×9−18×8)−2(12×9−18×7)+3(12×8−15×7)=0
Thus, the value of the determinant remains unchanged.
Example 2:
Consider the following determinant:
| 1 2 3 |
| 4 5 6 |
| 7 8 9 |
If the first row is multiplied by 2, the determinant becomes:
| 2 4 6 |
| 4 5 6 |
| 7 8 9 |
The value of the original determinant is:
1(5×9−6×8)−2(4×9−6×7)+3(4×8−5×7)=0
The value of the new determinant is:
2(5×9−6×8)−4(4×9−6×7)+6(4×8−5×7)=0
Thus, the value of the determinant remains unchanged.
When a row of a determinant is multiplied by a constant k, the determinant changes, but there is a simple relationship between the original determinant and the new one. Specifically, if the determinant is A and the new determinant is B, then B can be obtained from A by multiplying one row of A by k, as follows:
- If a row of A is multiplied by k, then the corresponding element of B is k times the original element.
- All other elements of B are the same as the corresponding elements of A.
- Thus, B is obtained by replacing one row of A by its multiple by k.
Effect of Multiplication of a Row by a Constant on the Value of a Determinant
The value of a determinant is a scalar quantity that depends on the elements of the determinant. When a row of a determinant is multiplied by a constant k, the value of the determinant changes by a factor of k. Specifically, if the determinant is A and the new determinant is B, then the value of B is k times the value of A.
Examples
Example 1:
Consider the following determinant:
| 1 2 3 |
| 4 5 6 |
| 7 8 9 |
If the second row is multiplied by 3, the determinant becomes:
| 1 2 3 |
| 12 15 18 |
| 7 8 9 |
The value of the original determinant is:
1(5×9−6×8)−2(4×9−6×7)+3(4×8−5×7)=0
The value of the new determinant is:
1(15×9−18×8)−2(12×9−18×7)+3(12×8−15×7)=0
Thus, the value of the determinant remains unchanged.
Example 2:
Consider the following determinant:
| 1 2 3 |
| 4 5 6 |
| 7 8 9 |
If the first row is multiplied by 2, the determinant becomes:
| 2 4 6 |
| 4 5 6 |
| 7 8 9 |
The value of the original determinant is:
1(5×9−6×8)−2(4×9−6×7)+3(4×8−5×7)=0
The value of the new determinant is:
2(5×9−6×8)−4(4×9−6×7)+6(4×8−5×7)=0
Thus, the value of the determinant remains unchanged.
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If each element in a row of a determinant is multiplied by the same fa...
Multiplying a Row by a Constant Factor in a Determinant
When we multiply a row in a determinant with a constant factor, it affects the value of the determinant. Let's see how:
- Consider a determinant A of order n.
- Let's say we multiply each element in a particular row (let's say row i) by a constant factor r.
- Then the new determinant A' will be obtained by replacing the ith row of A with the corresponding row obtained by multiplying each element by r.
Example:
Let's take an example determinant A:
| 1 2 3 |
| 4 5 6 |
| 7 8 9 |
Now, let's multiply the second row (row 2) by a constant r = 2. The new determinant A' will be:
| 1 2 3 |
| 8 10 12 |
| 7 8 9 |
Effect on the Value of Determinant
When we multiply a row in a determinant by a constant factor r, it affects the value of the determinant as follows:
- The value of the determinant is multiplied by r.
In other words, if the original value of the determinant was D, then the new value of the determinant will be rD.
This can be proved by expanding the determinant along the row that we multiplied by r. Let's take the example determinant A' above and expand it along the second row:
| 1 2 3 |
| 8 10 12 |
| 7 8 9 |
= 8 | 1 3 |
| 7 9 |
- 2 | 4 6 |
| 7 9 |
= 2(4*9 - 6*7) + 8(1*9 - 3*7)
= -12 + 24
= 12
Now, let's see what happens to the value of the determinant if we multiply the second row of A by a constant factor of r:
| 1 2 3 |
| 8r 10r 12r |
| 7 8 9 |
Expanding this determinant along the second row gives:
| 1 2 3 |
| 8r 10r 12r |
| 7 8 9 |
= 8r | 1 3 |
| 7 9 |
- 2 | 4 6 |
| 7 9 |
= 2(4*9 - 6*7) + 8r(1*9 - 3*7)
= -12 + 24r
= 12r
Thus, we see that the value of the determinant is multiplied by r. Therefore, the correct answer is option D: "is multiplied by r."
When we multiply a row in a determinant with a constant factor, it affects the value of the determinant. Let's see how:
- Consider a determinant A of order n.
- Let's say we multiply each element in a particular row (let's say row i) by a constant factor r.
- Then the new determinant A' will be obtained by replacing the ith row of A with the corresponding row obtained by multiplying each element by r.
Example:
Let's take an example determinant A:
| 1 2 3 |
| 4 5 6 |
| 7 8 9 |
Now, let's multiply the second row (row 2) by a constant r = 2. The new determinant A' will be:
| 1 2 3 |
| 8 10 12 |
| 7 8 9 |
Effect on the Value of Determinant
When we multiply a row in a determinant by a constant factor r, it affects the value of the determinant as follows:
- The value of the determinant is multiplied by r.
In other words, if the original value of the determinant was D, then the new value of the determinant will be rD.
This can be proved by expanding the determinant along the row that we multiplied by r. Let's take the example determinant A' above and expand it along the second row:
| 1 2 3 |
| 8 10 12 |
| 7 8 9 |
= 8 | 1 3 |
| 7 9 |
- 2 | 4 6 |
| 7 9 |
= 2(4*9 - 6*7) + 8(1*9 - 3*7)
= -12 + 24
= 12
Now, let's see what happens to the value of the determinant if we multiply the second row of A by a constant factor of r:
| 1 2 3 |
| 8r 10r 12r |
| 7 8 9 |
Expanding this determinant along the second row gives:
| 1 2 3 |
| 8r 10r 12r |
| 7 8 9 |
= 8r | 1 3 |
| 7 9 |
- 2 | 4 6 |
| 7 9 |
= 2(4*9 - 6*7) + 8r(1*9 - 3*7)
= -12 + 24r
= 12r
Thus, we see that the value of the determinant is multiplied by r. Therefore, the correct answer is option D: "is multiplied by r."
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If each element in a row of a determinant is multiplied by the same factor r, then the value of the determinanta)is multiplied by r3b)is increased by 3rc)remains unchangedd)is multiplied by rCorrect answer is option 'D'. Can you explain this answer?
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