3 x 3 determinant gives the volume of a parallelopiped

[ajayguhan]

I know that 3 x 3 determinant gives the volume of a parallelopiped, but how come after the row operations also it's gives the Same volume when it's elements are changed or in another words it's sides are being modified?

 

[SteamKing]

You'll have to be more specific in your description.

If you study the properties of determinants, you'll see that for certain row operations, the determinant isn't changed. This property comes in handy when trying to solve a linear system of equations.

http://en.wikipedia.org/wiki/Determinant

 

[ajayguhan]

In square matrix multiplication of 3 x3 . consider two matrix A, B such that AB =C ,to obtain the c11 element of C, we take a dot product of row 1 of A and column 1 of B. Row 1 of A is vector whose x, y, z components are a11, a12, a13 respectively. But column 1 of B consist of only x component of three vector of B and I'm taking dot product of a vector and x components to get single element or x component of single vector in C.

Note each matrix A and B consist of 3 different vectors specifying a parallelopiped and x, y, z components are written in column 1,2,3 respectively. Det of A and B is non zero

My question how does the dot product of row 1 and column 1 gives the x component of vector in C is there any proof?



Thanks in advance

 

[HallsofIvy]

You seem to be under the impression that if matrix "A" has determinant d, then matrix B, derived from A by row operations, has the same determinant. That is NOT true.

For example, swapping two rows multiplies the determinant by -1. Multiplying a row by "a" multiplies the determinant by "a". It is true that "adding a multiple of one row to another" does not change the determinant.

 

[blue_raver22]

I can explain this phenomenon using the principles of linear algebra. The determinant of a matrix is a scalar value that represents the volume of the parallelepiped formed by the column vectors of the matrix. This means that the determinant is a measure of the size and orientation of the parallelepiped.

When performing row operations on a matrix, we are essentially changing the orientation and size of the parallelepiped. However, these operations do not change the overall volume of the parallelepiped. This is because the determinant is a measure of the relative change in size and orientation, not the absolute values. So while the individual elements may change, the overall volume remains the same.

In other words, the determinant is a measure of the transformation of the parallelepiped, not the specific values of its sides. This is why even after row operations, the determinant still gives the same volume of the parallelepiped.