Determining Values for a Given Matrix - How Can I Simplify This Process?

In summary, the determinant of matrix A is equal to 0. By using the last two rows, it can be determined that x must be equal to -1 for detA = 0. However, the correct answer given is x = -1 or 3. The determinant can be calculated using the expansion by rows trick, but it can also get messy. It is recommended to calculate the determinant and solve the polynomial to find the roots.
  • #1
dracolnyte
28
0

Homework Statement


detA = 0
Matrix A =

| (x+5) 4 4 |
| -4 (x-3) -4 |
| -4 -4 (x-3)|

The Attempt at a Solution



I know that if one row or column is equal to another, then detA = 0, so using the last 2 rows, i can find out that x has to be -1 for row2 and row3 to be equal for detA = 0.

but the answer at the back says x = -1 or 3, how can i solve the 3? I have tried to reduce it to the triangular form, but it got way too messy to be correct. I also tried using the 3x3 matrix trick where you copy the first 2 rows and make a 4th and 5th row out of them and solve for the determinant, also got pretty messy.

Is there some rule that i missed out that can make my life easier on solving this question?
 
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  • #2


Do you know the expansion by rows trick?

Definitely the best way to do this is to calculate the determinant and then solve the polynomial... trying to find linear dependence conditions is too error prone and you can miss solutions
 
  • #3


you mean the one the one where you the multiply the diagonals and the subtract it by the other diagonals? in other words
(x+5) 4 4
-4 (x-3) -4
-4 -4 (x-3)
------------
(x+5) 4 4
-4 (x-3) -4
 
  • #4


a11a22a33 - a11a23a32 - a12a21a33 + a12a23a31 + a13a21a32 - a13a22a31 = 0
it gets really messy with like = x^3 - 11x^2 + 55x - 93
 
Last edited:
  • #5


dracolnyte said:
a11a22a33 - a11a23a32 - a12a21a33 + a12a23a31 + a13a21a32 - a13a22a31 = 0
Either your formula isn't right, or you have made in error in calculation.
I worked it out and got a different polynomial, which when factored and set to zero, had roots equal to 3 and -1.
dracolnyte said:
it gets really messy with like = x^3 - 11x^2 + 55x - 93
 
  • #6


ya sorry, i realized and i did it again, i got x = 3 and -1. my bad, it must be getting late
 

1. What are determinants in math?

Determinants are mathematical objects that are used to solve systems of linear equations and to study properties of linear transformations. They are typically represented by a square matrix and are used to calculate the volume or area of a geometric object.

2. How do I find the determinant of a matrix?

To find the determinant of a matrix, you can use the cofactor expansion method or the row reduction method. The cofactor expansion method involves multiplying each element in a row or column by its corresponding minor (determinant of the smaller matrix formed by removing the row and column of that element) and adding these products together. The row reduction method involves using elementary row operations to transform the matrix into an upper or lower triangular matrix, and then taking the product of the elements on the main diagonal.

3. What is the importance of determinants in linear algebra?

Determinants are important in linear algebra because they provide a way to determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions. They are also used to find the inverse of a matrix and to calculate the area or volume of geometric objects. Additionally, determinants are used to study properties of linear transformations, such as whether they are invertible or preserve orientation.

4. Can determinants be negative?

Yes, determinants can be negative. The sign of the determinant depends on the order of the rows and columns in the matrix. If the rows are switched, the sign of the determinant will change. If the columns are switched, the sign will also change. However, the magnitude of the determinant will remain the same.

5. Are there any special properties of determinants?

Yes, there are several special properties of determinants that are useful in solving systems of linear equations and in other applications. These include the fact that the determinant of a triangular matrix is equal to the product of the elements on the main diagonal, the determinant of a matrix multiplied by a scalar is equal to the scalar multiplied by the determinant, and the determinant of the identity matrix is equal to 1.

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