Linear Algebra Matrix with Elementary Row Operations

In summary, the 3x3 matrix A is transformed into the identity matrix I by five elementary row operations. The final determinant of A is -1/4, found by working backwards through the row operations and taking into account the effect on the determinant. The determinant of the starting matrix A can also be found by selecting the row operations that affect the determinant and using the determinant of the identity matrix (1) to find the determinant of A.
  • #1
lina29
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Homework Statement



The 3x3 matrix A is transformed into I by the following elementary row operations
R1+2R3 -> R1
R2+2R3 ->R2
2R2 ->R2
R1 <->R2
2R3 ->R3

Find det(A)

Homework Equations



I assumed to start off with the problem since I was going backwards from I to A. I would do the opposite of each row operation ie
2R3-R1 ->R1
2R3-R2 ->R2
(1/2)R2 ->R2
R1 <->R2
(1/2)R3 ->R3

The Attempt at a Solution



By finding the det(A) I got -1/4. I'm confused on if I messed up on the row operations. When I did this problem without using the backwards row operations I got -4 which was also wrong. I'd appreciate any help

Thanks
 
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  • #2
Of the three elementary row operations, only one of them changes the value of the determinant of the matrix. Do you know which one this is?

Since you end up with the identity matrix (det(I) = 1), you can pick out the row operations that affect the determinant, to get the determinant of your starting matrix.
 

1. What are elementary row operations in linear algebra?

Elementary row operations in linear algebra refer to the three basic operations that can be performed on a matrix to transform it into an equivalent matrix: multiplying a row by a non-zero constant, interchanging two rows, and adding a multiple of one row to another row.

2. How do elementary row operations affect the determinant of a matrix?

Elementary row operations do not affect the determinant of a matrix. The determinant remains the same as long as the operations are performed on the same matrix. This is because the determinant is a property of the matrix itself, not the operations performed on it.

3. Can elementary row operations change the rank of a matrix?

Yes, elementary row operations can change the rank of a matrix. The rank of a matrix is the number of linearly independent rows or columns in the matrix. By performing elementary row operations, we can transform a matrix into different forms and potentially change the number of linearly independent rows or columns, thus changing the rank.

4. How do elementary row operations help in solving systems of linear equations?

Elementary row operations are essential in solving systems of linear equations using matrices. By performing these operations, we can transform a system of equations into an equivalent one that is easier to solve. This is because the operations help to eliminate variables and simplify the equations, making it easier to find the solution.

5. Is it always possible to find an inverse of a matrix using elementary row operations?

No, it is not always possible to find an inverse of a matrix using elementary row operations. A matrix must be square and have a non-zero determinant to have an inverse. If a matrix does not meet these criteria, an inverse cannot be found using elementary row operations.

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