# Linear algebra : calculating the determinant

• fluidistic
In summary: Oh well, that's maybe a bit too late for me right now. I just saw my error! Unforgivable, I wrote that (det(A)^2)^3=det(A)^5 instead of det(A)^6. I got it now. Thank you!
fluidistic
Gold Member
-1. Homework Statement
Let A and B be 2x2 matrices such that A²B=3I and A^T*B³=-I, where A^T is the transposed matrix of A, and I is the identity matrix.
Calculate det(A).
0. The attempt at a solution
I know I don't know how to approach correctly the problem, but I've tried something.
I know that the determinant of A squared times the determinant of B is equal to 9 and that minus the determinant of A times the determinant of B cubed is worth 1. From this, I can't reach the answer. And I've no idea about a different approach.
If you have an idea, I'd be glad to be its tester.

Last edited:
What exactly is the question?

gabbagabbahey said:
What exactly is the question?

Oops you're right. The question is to find out det (A).
I edit the first post to make it clearer.

Okay, in that case you are on the right track ; basically you have 2 equations:

$$(\det A)^2\det B=9$$ and $$(\det A)(\det B)^3=1$$

with two unknowns; $\det A$ and $\det B$

How would you normally go about solving a system of two (non-linear) equations and two unknowns? Try solving the first for $\det B$ and substituting it into the second.

gabbagabbahey said:
Okay, in that case you are on the right track ; basically you have 2 equations:

$$(\det A)^2\det B=9$$ and $$(\det A)(\det B)^3=1$$

with two unknowns; $\det A$ and $\det B$

How would you normally go about solving a system of two (non-linear) equations and two unknowns? Try solving the first for $\det B$ and substituting it into the second.
That's what I've tried to do, but I gave up because I made an error.
Well now you gave me confidence and I could reach a result : det (A)=$$\sqrt [4] {9^3}$$. Thank you very much!

fluidistic said:
That's what I've tried to do, but I gave up because I made an error.
Well now you gave me confidence and I could reach a result : det (A)=$$\sqrt [4] {9^3}$$. Thank you very much!

Unfortunately you still have an error; you should get $$\det A =9^{3/5}$$

gabbagabbahey said:
Unfortunately you still have an error; you should get $$\det A =9^{3/5}$$

Oh well, that's maybe a bit too late for me right now. I just saw my error! Unforgivable, I wrote that $$(det(A)^2)^3=det(A)^5$$ instead of $$det(A)^6$$.
I got it now. Thank you!

## 1. What is the definition of a determinant in linear algebra?

The determinant of a matrix is a numerical value that can be calculated from the elements of the matrix. It is a property of square matrices and is denoted by either det(A) or |A|. It represents the scaling factor of the transformation described by the matrix.

## 2. How is the determinant calculated for a 2x2 matrix?

The determinant of a 2x2 matrix can be calculated using the following formula:
|A| = ad - bc
Where a, b, c, and d are the elements of the matrix arranged in a 2x2 grid.

## 3. What is the importance of the determinant in linear algebra?

The determinant is an important tool in linear algebra as it allows us to determine if a matrix is invertible, and if so, the magnitude of the inverse. It is also used to solve systems of linear equations and determine the area and volume of geometric objects in higher dimensions.

## 4. Can the determinant be negative?

Yes, the determinant can be negative. If the determinant is negative, it means that the transformation described by the matrix includes a reflection or a combination of reflections. If the determinant is positive, it means that the transformation includes only rotations and/or translations.

## 5. How does changing the elements of a matrix affect its determinant?

Changing the elements of a matrix can have different effects on its determinant. For example, multiplying a row (or column) of a matrix by a constant will result in the determinant being multiplied by the same constant. Swapping two rows (or columns) will result in the determinant being multiplied by -1. Adding or subtracting a multiple of one row (or column) to another will not change the determinant. These properties of determinants are useful in calculating the determinant of larger matrices.

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