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How to Calculate Volume Using a Matrix

The concept of finding volume using a matrix is usually taught in a linear algebra course. To find volume, the matrix is made from vectors put into columns. The matrix A for a parallelepiped made from the vectors

[x1, y1, z1], [x2, y2, z2], and [x3, y3, z3] is represented by:

[ x1 x2 x3

y1 y2 y3 = A

z1 z2 z3]

The concept of volume in linear algebra has a different meaning than in a geometry course. Volume will represent the size of the figure represented by the number of vectors in the matrix. For instance, a matrix with one vector, 1-box, has a volume that represents its length. The graph is one vector. A matrix with two vectors, 2-box, has a volume that represents its area. The graph is a parallelogram. A matrix with three vectors, 3-box, has the volume representing the space inside. The graph is a parallelepiped. More dimensions, n, can be added beyond three, n-box, but formal names are not given. The volume of the n-box in an m by n matrix is found by taking the square root of the determinant of the product of the matrix transpose and matrix, V = sqrt (det(A^T*A). In a special case of an n by n matrix, the volume is just the absolute value of the determinant of A, V = abs (det(A)).

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Instructions

Finding Volume of an N-box Formed from an M by N Vector Matrix
- 1
  Set up the vector matrix, A. Example: Find the volume of the parallelepiped determined by the vectors [2, 3, -1], [-4, 5, 0], [1, -2, 4].
  
  [2 | -4 | 1
  
  3 | 5 | -2
  
  -1 | 0 | 4]
- 2
  Transpose matrix A to find A^T. The rows of A become the columns of A^T.
  
  [2 | 3 | -1
  
  -4 | 5 | 0
  
  1 | -2 | 4]
- 3
  Multiply A^T and A together using rules for matrix multiplication. Row 1 in A^T multiples to column 1 in A with each multiply step being summed to find the new 1,1 value. 2(2)+3(-4)+(-1)(1) = 4-12-1 = -9. Row 1 in A^T multiples to column 2 in A with each multiply step being summed to find the new 1,2 value. 2(-4)+3(5)+(-1)(0) = -8+15+0 = 7. The process continues until all 9 values are found in the multiplication matrix.
  
  [2(2)+3(3)+(-1)(-1) | 2(-4)+3(5)+(-1)(0) | 2(1)+3(-2)+(-1)(4)
  
  -4(2)+5(3)+0(-1) | (-4)(-4)+5(5)+0(0) | (-4)(1)+5(-2)+0(4)
  
  1(2)+(-2)(3)+4(-1) | 1(-4)+(-2)(5)+4(0) | 1(1)+(-2)(-2)+4(4)]
  
  [14 | 7 | -8
  
  7 | 41 | -14
  
  -8 | -14 | 21]
- 4
  Calculate the value of the determinant for the product matrix, A^T*A.
  
  14(41)(21)+7(-14)(-8)+(-8)(7)(-14)-[(-8)(41)(-8)+14(-14)(-14)+7(7)(21)]
  
  12054+784+784-[22624+2744+1029]
  
  13622-6397
  
  7225
- 5
  Calculate the square root of the result to find the volume. The volume of the example is the square root of 7225. The volume result is 85 cubic units.
Finding Volume of an N-box Formed from an N by N Square Vector Matrix
- 6
  Set up the vector matrix, A. Example: Find the volume of the parallelepiped determined by the vectors [2, 3, -1], [-4, 5, 0], [1, -2, 4].
  
  [2 | -4 | 1
  
  3 | 5 | -2
  
  -1 | 0 | 4]
- 7
  Calculate the determinant of A.
  
  2(5)(4)+(-4)(-2)(-1)+1(3)(0)-[1(5)(-1)+2(-2)(0)+(-4)(3)(4)]
  
  40-8+0-[-5+0-48]
  
  32-(-53)
  
  85
- 8
  Find the absolute value of the result. The absolute value of 85 is 85.
  
  The volume of the parallelepiped is 85 cubic units.

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