3 x 3 determinant gives the volume of a parallelopiped

[ajayguhan]

Main Question or Discussion Point

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.