Statement on matrix and determinant

1. Apr 27, 2015

Raghav Gupta

1. The problem statement, all variables and given/known data
If A is a square matrix of order 3 then the true statement is
1. det(-A) = - det A
2.det A = 0
3.det ( A + I) = I + detA
4.det(2A) = 2detA
2. Relevant equations
NA

3. The attempt at a solution
2. option is obviously not true.
Making a random matrix A and verifying properties 1. , 3. , 4. would be lengthy.
What is the approach for this?

2. Apr 27, 2015

Orodruin

Staff Emeritus
Did you try simply using some basic rules for determinants? How would you write the determinant of a matrix in terms of its components?

3. Apr 27, 2015

Raghav Gupta

I know det(AT) = det(A)
for equal size square matrices,
det(A)(B) = det(A)det(B),
What property should I apply as these I think are not applicable here?

4. Apr 27, 2015

Orodruin

Staff Emeritus
Can you write multiplication by a constant as a multiplication with a matrix? In that case, what matrix?

5. Apr 27, 2015

Raghav Gupta

Is it the identity matrix $I$ ?

6. Apr 27, 2015

PeroK

Why not take $A = I$, and calculate the determinants required? Just to see what happens. You are allowed to try things out with specific matrices!

7. Apr 27, 2015

Orodruin

Staff Emeritus
Do you get 3A by multiplying A with the identity?

8. Apr 27, 2015

Raghav Gupta

No. We get A only.

9. Apr 27, 2015

Orodruin

Staff Emeritus
So how would you get 3A?

10. Apr 28, 2015

Raghav Gupta

By multiplying A with 3 or 3$I$

11. Apr 28, 2015

Orodruin

Staff Emeritus
Yes, try the latter and apply your determinant relations.

12. Apr 28, 2015

Raghav Gupta

But by that I am getting options 1 and 4 true.
Among that one is supposed to be true.

13. Apr 28, 2015

Orodruin

Staff Emeritus
You are then doing it wrong. What is the determinant of 3I?

14. Apr 28, 2015

Raghav Gupta

27

15. Apr 28, 2015

Orodruin

Staff Emeritus
Yes, so what is then the determinant of 2A expressed in det(A)?

16. Apr 28, 2015

Raghav Gupta

8det(A).Option 4 rejected.
Option 1 is okay.

17. Apr 28, 2015

Ray Vickson

You tell us.

18. Apr 28, 2015

Raghav Gupta

But we can't apply the property there of
det(A)det(B) = det(A)(B) as there is no use of it?

19. Apr 28, 2015

Orodruin

Staff Emeritus
Why dont you try insering a well chosen matrix and see what comes out? I suggest one which will make the determinants easy to compute.

20. Apr 28, 2015

Raghav Gupta

Oh, thanks that option 3 is also false.
But I have verified that by taking any random matrix.
Is there any general proof of that or we learn it in higher sections?