Statement on matrix and determinant

In summary: I always took the I there to mean 1, precisely since it does not have any meaning otherwise ...In some contexts (especially in books on functional analysis) it's considered acceptable to write down equations of the type "operator = number" and sums of the type "operator + number". The number is then interpreted as that number times the identity operator. So (3) could be interpreted as ##\det(A+I)I=I+(\det A)I##, but it seems very unlikely that this is the intended interpretation. I'm inclined to go with Ray Vickson's interpretation first (the equality is nonsense), and Orodruin's interpretation second (the I on the right is supposed to be 1).
  • #1
Raghav Gupta
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Homework Statement


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

Homework Equations


NA

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?
 
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  • #2
Raghav Gupta said:
What is the approach for this?

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
Orodruin said:
Did you try simply using some basic rules for determinants? How would you write the determinant of a matrix in terms of its components?
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
Raghav Gupta said:
det(A)(B) = det(A)det(B),

Can you write multiplication by a constant as a multiplication with a matrix? In that case, what matrix?
 
  • #5
Orodruin said:
Can you write multiplication by a constant as a multiplication with a matrix? In that case, what matrix?
Is it the identity matrix ## I ## ?
 
  • #6
Raghav Gupta said:
Is it the identity matrix ## I ## ?

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
Do you get 3A by multiplying A with the identity?
 
  • #8
Orodruin said:
Do you get 3A by multiplying A with the identity?
No. We get A only.
 
  • #9
So how would you get 3A?
 
  • #10
Orodruin said:
So how would you get 3A?
By multiplying A with 3 or 3##I##
 
  • #11
Raghav Gupta said:
By multiplying A with 3 or 3##I##
Yes, try the latter and apply your determinant relations.
 
  • #12
Orodruin said:
Yes, try the latter and apply your determinant relations.
But by that I am getting options 1 and 4 true.
Among that one is supposed to be true.
 
  • #13
You are then doing it wrong. What is the determinant of 3I?
 
  • #14
Orodruin said:
You are then doing it wrong. What is the determinant of 3I?
27
 
  • #15
Raghav Gupta said:
27
Yes, so what is then the determinant of 2A expressed in det(A)?
 
  • #16
Orodruin said:
Yes, so what is then the determinant of 2A expressed in det(A)?
8det(A).Option 4 rejected.
Option 1 is okay.
What about option 3 ?
 
  • #17
Raghav Gupta said:
8det(A).Option 4 rejected.
Option 1 is okay.
What about option 3 ?

You tell us.
 
  • #18
Ray Vickson said:
You tell us.
But we can't apply the property there of
det(A)det(B) = det(A)(B) as there is no use of it?
 
  • #19
Why don't you try insering a well chosen matrix and see what comes out? I suggest one which will make the determinants easy to compute.
 
  • #20
Orodruin said:
Why don't you try insering a well chosen matrix and see what comes out? I suggest one which will make the determinants easy to compute.
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?
 
  • #21
To prove that something is false you only need a counter example.
 
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  • #22
Raghav Gupta said:
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?

Right: not only is (3) false, it does not even have any meaning. The left-hand-side det(A+I) is a number, while the right-hand-side det(A) + I is a number plus a matrix---which does not exist in any reasonable way.
 
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  • #23
Ray Vickson said:
Right: not only is (3) false, it does not even have any meaning. The left-hand-side det(A+I) is a number, while the right-hand-side det(A) + I is a number plus a matrix---which does not exist in any reasonable way.
I always took the I there to mean 1, precisely since it does not have any meaning otherwise ...
 
  • #24
In some contexts (especially in books on functional analysis) it's considered acceptable to write down equations of the type "operator = number" and sums of the type "operator + number". The number is then interpreted as that number times the identity operator. So (3) could be interpreted as ##\det(A+I)I=I+(\det A)I##, but it seems very unlikely that this is the intended interpretation. I'm inclined to go with Ray Vickson's interpretation first (the equality is nonsense), and Orodruin's interpretation second (the I on the right is supposed to be 1).
 

Related to Statement on matrix and determinant

What is a matrix and determinant?

A matrix is a rectangular array of numbers or variables arranged in rows and columns. A determinant is a numerical value that can be calculated using the values in a matrix.

What is the purpose of a matrix and determinant in science?

Matrices and determinants are used in various fields of science, such as physics, engineering, and computer science, to represent and solve systems of equations, perform transformations, and analyze data.

How do you calculate the determinant of a matrix?

The determinant of a matrix can be calculated by following a specific formula depending on the size of the matrix. For a 2x2 matrix, the determinant is found by multiplying the values in the main diagonal and subtracting the product of the values in the other diagonal. For larger matrices, the process involves expanding the matrix into smaller submatrices until a 2x2 matrix is reached.

What is the relationship between a matrix and its determinant?

The determinant of a matrix is a unique value that is determined by the values in the matrix. It provides information about the properties of the matrix, such as its invertibility and the volume of parallelepiped formed by the column vectors of the matrix.

Can a matrix have more than one determinant?

No, a matrix can only have one determinant. The determinant of a matrix is a single value that is determined by the values in the matrix. Changing any value in the matrix will result in a different determinant.

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