Solving for the Determinent of a Matrix

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In summary, the determinant function is defined only for square matrices and cannot be calculated for non-square matrices. The determinant characterizes when a matrix has an inverse, and only square matrices can have inverses. The cross product can be used to find the determinant of a 3x3 matrix, but not for non-square matrices. It is possible to compute all determinants of square submatrices, but the determinant of a non-square matrix is either not defined or zero.
  • #1
Poweranimals
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Okay, I'm learning currently how to solve for the determinent of a Matrix. Of course the book explains how to solve for a 2 X 2 Matrix, a 3 X 3 Matrix, a 4 X 4 Matrix, ect. But it says nothing about how to solve for a 3 X 2 Matrix.

Any idea how to do this? I'm really baffled on this.
 
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  • #2
Just in case no one knows what I'm talking about, by a 3 X 2 Matrix, I mean like this:

[1 0 -2]
[2 1 -1]

It's very hard to figure this out. My book doesn't even go over it. They'll go over 2 X 2, 3 X 3, 4 X 4, 5 X 5, 6 X 6, ect.. But for some reason they don't touch on if the Matrix doesn't have equal sides.
 
  • #3
But it says nothing about how to solve for a 3 X 2 Matrix.

No wonder, the determinant function (or "a determinantal function") is defined as a function from the set of all nxn (i.e square) matrices (with elements in a field F), to the field F (the determinant takes a square matrix and spits back out a number). There are reasons for this.

"The" inverse of a matrix A is a matrix B such that AB = BA = I (i.e. B is both a left inverse and a right inverse of A). Suppose A is of size nxm, and B is pxq. Then AB is nxq and BA is pxm. But I is square, say I is a txt matrix. Since AB = BA = I, this forces n = q = p = m = t, i.e. A and B are both square. Thus only square matrices can have inverses.

One wants the determinant function to characterize when exactly a matrix X has an inverse (it just so happens to be that X has an inverse iff det(X) != 0). But according to the above, only square matrices can have inverses, so it doesn't make much sense to define the determinant for non-square matrices.

Btw, a 3x2 matrix has 3 rows and 2 columns. The matrix you used as an example is a 2x3 matrix.
 
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  • #4
So is it possible to solve the determinent of non square Matrices? Or is the answer simply Cannot be calculated?
 
  • #5
No, using the ordinary definition of the determinant function, you cannot calculate the determinant of a non-square matrix.
 
  • #6
It's not so much that you "cannot calculate" it as that it is not defined!

The determinant is only defined for square matrices.

Asking how to find the determinant of a 3 by 2 matrix is a lot like asking how to find the square root of a chair.
 
  • #7
you can solve a 2x3 matrix, you use this all the time to when multiplying vectors.

[a b c]
[e f g]

(bg-cf)-(ag-ce)+(af-be)
 
  • #8
korciuch said:
you can solve a 2x3 matrix, you use this all the time to when multiplying vectors.

[a b c]
[e f g]

(bg-cf)-(ag-ce)+(af-be)

So, this is called the korciuch-operator? :biggrin:
 
  • #9
The phrase "solve a matrix" doesn't make sense. The orginal question was "solve for the determinant of a matrix". Again, a 3 by 2 matrix (or any non-square matrix) does not have a determinant.
 
  • #10
korciuch said:
you can solve a 2x3 matrix, you use this all the time to when multiplying vectors.

[a b c]
[e f g]

(bg-cf)-(ag-ce)+(af-be)

thats the cross product, which in this case, is a 3x3 matrix with [i j k] as the first row:
[i j k]
[a b c]
[e f g]
 
  • #11
the det of a non square matrix is either not defined or zero.
 
  • #12
Could you give an example of a non-square matrix that has determanant 0?
 
  • #13
You could try the products of the nonzero singular values in the singular value decomposition. It's not the determinant, but it's the closest thing you're going to get.
 
  • #14
I was wondering why mathwonk said
the det of a non square matrix is either not defined or zero.
 
  • #15
I was wondering the same thing too, when I came across his post. I don't think the guru would have made a mistake over something like this. So I keep wondering...

Edit: Googling does throw up some links about non square determinants. I still can't understand the motivation to assign a 0 value to a non square det.
 
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  • #16
mathwonk is right. You can't get the determinant of a non square matrix. The determinant of a matrix is the n dimensional volume spanned by n vectors. If you don't have n vectors this value either can't be obtained, or is zero.

I believe there is a way of asking for the k<n dimensional volume of k, n-dimensional, vectors, but I don't know how one would go about this. You could try a change of coordinates , via the singular value decomposition say. I believe the products of the non zero singular values do give you the k dimensional volume spanned by k column vectors. I don't have any references for this, but I'm pretty sure it's ok.
 
  • #17
of course you can compute all the determinants of square sub matrices.

e.g to coordinatize planes in 4 space, you can consider them as sappned by the columns of 4x2 matrices, and compute all, let's see, 4 choose 2 is 6, 2x2 subdeterminants.

this gives projective coordinates on the grassmannian of 2 plkanes in 4 space, embedding it in projective 5 space, since changing the basis for a plane changes those determinants all by the same scalar factor.

i.e. essentially yiu are forming a parallelogram in your plane, then taking all 6 projections of that parallelogram onto pairs of axes, and computing their oriented areas.

by the pythagorean theorem for areas, this also allows computation of the area of the original parallelogram in 4 space.
 

What is the Determinant of a Matrix?

The determinant of a square matrix is a scalar value that can be computed from the elements of the matrix. It is a fundamental concept in linear algebra and is used to determine various properties of the matrix, such as its invertibility and its effect on linear transformations.

How is the Determinant of a Matrix Noted?

The determinant of a matrix is typically denoted using vertical bars around the matrix, like this: |A|, where A is the matrix. If A is a square matrix, |A| represents its determinant.

How is the Determinant of a 2x2 Matrix Calculated?

For a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), the determinant is calculated using the formula:

|A| = ad - bc

How is the Determinant of a 3x3 Matrix Calculated?

For a 3x3 matrix \(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\), the determinant is calculated using the formula:

|A| = a(ei - fh) - b(di - fg) + c(dh - eg)

Are There Any Properties of Determinants?

Yes, determinants have several properties, including:

  • 1. The determinant of the identity matrix I is 1: |I| = 1.
  • 2. The determinant of a matrix and its transpose are the same: |A| = |Aᵀ|.
  • 3. The determinant of a matrix multiplied by a scalar k is k times the determinant of the original matrix: |kA| = k|A|.
  • 4. The determinant of the product of two matrices is the product of their determinants: |AB| = |A| * |B|.
  • 5. If two rows or columns of a matrix are exchanged, the sign of the determinant changes.

How Can I Calculate the Determinant of a Larger Matrix (4x4 or Higher)?

For larger matrices, you can calculate the determinant using various methods, including expansion by minors, row reduction, or using software tools like calculators or computer software with linear algebra capabilities.

What Does the Determinant Tell Us About a Matrix?

The determinant of a matrix provides valuable information about the matrix's properties, such as:

  • 1. Whether the matrix is invertible (non-singular): If |A| ≠ 0, then A is invertible.
  • 2. The volume scaling factor: The determinant of a matrix can indicate how it scales the volume of a shape in geometric transformations.
  • 3. Orientation preservation: A positive determinant implies that the matrix preserves the orientation of vectors, while a negative determinant reverses it.

Why is Calculating the Determinant Important?

Calculating the determinant is essential in various fields, including linear algebra, calculus, physics, and computer science. It is used to determine the solvability of systems of linear equations, find eigenvalues and eigenvectors, analyze transformations, and assess the invertibility of matrices, among other applications.

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