# Linear Algebra, scalars to find det(A)

• concon
In summary: Hey I actually just figured out how to solve it by using row operations and linear dependence and I got zero as the determinant. Is this correct?
concon

## Homework Statement

Let X1, X2,...,Xn be scalars. Calculate det(A) where
A= nxn matrix with [ x1+1 x1+2...x1+n
x2+1 x2+2...x2+n
... ... ...
xn+1 xn+2...xn+n]

## Homework Equations

det(A) = (aij)(-1)^(i+j)det(aij)

## The Attempt at a Solution

I have no clue how to even begin solving this problem I tried using above formula to no avail.

Spacing on matrix got messed up so imagine it with the rows lined up correctly

Review what you know about how determinants, row (or column) operations on the matrix, and linear dependence.

Hint: the answer is very simple, when n > 2.

Can you explain a little more about what operations you are talking about? I am really confused!

Have you tried computing the determinant for small n? Such as a 2x2, 3x3, 4x4. I'd recommend doing that.

kduna said:
Have you tried computing the determinant for small n? Such as a 2x2, 3x3, 4x4. I'd recommend doing that.

How would that help me calculate the determinant when n is unknown?

concon said:
How would that help me calculate the determinant when n is unknown?

Start with simple cases of, say, n = 2, 3 and maybe 4, just to see what is going on. Then do it for general n.

concon said:
How would that help me calculate the determinant when n is unknown?

The answer doesn't depend on n, except that n = 1 and n = 2 are special cases.

I don't know how to give you any more help than "think about row and column operations on the matrix, and linear dependence" without telling you the answer.

Writing out the matrix in full, for n = 3 or n = 4, might help.

If the only thing you know about determinants is your "relevant equation"
det(A) = (aij)(-1)^(i+j)det(aij)
I think you missed a lot of stuff in class, or you haven't read your textbook.

kduna said:
Have you tried computing the determinant for small n? Such as a 2x2, 3x3, 4x4. I'd recommend doing that.

Yes I did computr for 2 by 2 and 3 by 3, but I do not see a relationship between the determinants. How do I calculate det(A) for nxn?

Ray Vickson said:
Start with simple cases of, say, n = 2, 3 and maybe 4, just to see what is going on. Then do it for general n.

Okay I did determinant calculation for 2x2 and 3x3. How do I calculate for nxn? Please help!

Two hints:
(1) Why is $\det A = B(x_2, \dots, x_n)x_1 + C(x_2, \dots, x_n)$ for some functions $B$ and $C$?
(2) What is the determinant of a matrix with two identical rows?

pasmith said:
Two hints:
(1) Why is $\det A = B(x_2, \dots, x_n)x_1 + C(x_2, \dots, x_n)$ for some functions $B$ and $C$?
(2) What is the determinant of a matrix with two identical rows?

1. I have no clue

2. determinant would be zero. Is the answer zero?

concon said:
Yes I did computr for 2 by 2 and 3 by 3, but I do not see a relationship between the determinants. How do I calculate det(A) for nxn?

So, what did you get for n = 2 and for n = 3?

Ray Vickson said:
So, what did you get for n = 2 and for n = 3?
For n=2 det(A)= (x1 +1)(x2 +2) - (x1 +2)(x2 +1)
How does that correlate to unknown n?

concon said:
For n=2 det(A)= (x1 +1)(x2 +2) - (x1 +2)(x2 +1)
How does that correlate to unknown n?

What do you get for n = 3? I mean, expand out everything and simplify it down to as small an expression as you can get. I am 100% serious. Looking at just n = 2 is not enough to reveal the pattern.

pasmith said:
(2) What is the determinant of a matrix with two identical rows?

concon said:
2. determinant would be zero.

Can you think of a way to do row or column operations on the matrix, to make two rows identical, and not change the determinant?

AlephZero said:
Can you think of a way to do row or column operations on the matrix, to make two rows identical, and not change the determinant?
Hey I actually just figured out how to solve it by using row operations and linear dependence and I got zero as the determinant. Is this correct?

## 1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with linear equations, vectors, and matrices. It is used to study and solve systems of linear equations, as well as analyze geometric transformations.

## 2. What are scalars in Linear Algebra?

Scalars in Linear Algebra refer to single numbers or coefficients that are multiplied by vectors or matrices. They are used to scale or change the magnitude of a vector or matrix.

## 3. How do you find the determinant of a matrix?

The determinant of a matrix is a value that is calculated using the elements of the matrix. To find the determinant, you can use various methods such as the cofactor expansion method or the row reduction method.

## 4. What is the significance of finding the determinant of a matrix?

The determinant of a matrix is used to determine if a system of linear equations has a unique solution, infinite solutions, or no solution at all. It is also useful in calculating the inverse of a matrix and in solving systems of differential equations.

## 5. Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative, positive, or zero. The sign of the determinant depends on the arrangement of the elements in the matrix and can be used to determine the orientation of a geometric transformation.

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