# Solving for the Determinent of a Matrix

by Poweranimals
Tags: determinent, matrix, solving
 P: 68 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.
 P: 68 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.
P: 696
 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.

P: 68

## Solving for the Determinent of a Matrix

So is it possible to solve the determinent of non square Matrices? Or is the answer simply Cannot be calculated?
 P: 696 No, using the ordinary definition of the determinant function, you cannot calculate the determinant of a non-square matrix.
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,904 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.
 P: 1 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)
HW Helper
P: 3,225
 Quote by korciuch 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?
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,904 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.
P: 356
 Quote by korciuch 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]
 Sci Advisor HW Helper P: 9,428 the det of a non square matrix is either not defined or zero.
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,904 Could you give an example of a non-square matrix that has determanant 0?
 P: 406 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.
Math
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