Question

Can someone please explain how to tell if a matrix has an inverse?

Answer 1

Sure Ma'am

Answer 2

I think it's true because someone once said something about it having fractions?

Answer 3

Let A be a square n by n matrix over a field K (for example the field R of real numbers). The following statements are equivalent, ie, for any given matrix they are either all true or all false:

A is invertible, i.e. A has an inverse, is nonsingular, or is nondegenerate.
A is row-equivalent to the n-by-n identity matrix In.
A is column-equivalent to the n-by-n identity matrix In.
A has n pivot positions.
det A ≠ 0. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.
A has full rank; that is, rank A = n.
The equation Ax = 0 has only the trivial solution x = 0
Null A = {0}
The equation Ax = b has exactly one solution for each b in Kn.
The columns of A are linearly independent.
The columns of A span Kn
Col A = Kn
The columns of A form a basis of Kn.
The linear transformation mapping x to Ax is a bijection from Kn to Kn.
There is an n by n matrix B such that AB = In = BA.
The transpose AT is an invertible matrix (hence rows of A are linearly independent, span Kn, and form a basis of Kn).
The number 0 is not an eigenvalue of A.
The matrix A can be expressed as a finite product of elementary matrices.
The matrix A has a left inverse (i.e. there exists a B such that BA = I) or a right inverse (i.e. there exists a C such that AC = I), in which case both left and right inverses exist and B = C = A-1.

Answer 4

if any one of these things are true then they are all true.

Answer 5

A matrix A has an inverse if and only detA is nonzero. So if the determinant is zero, it doesn't have an inverse. For a 2x2 matrix
[a b]
[c d],
The determinant is ad - bc. So if ad - bc = 0, there is no inverse.

The reason the determinant must be nonzero for there to be an inverse is, the formula for inverse of a 2x2 matrix is 1/(ad-bc)*A, where A is a 2x2 matrix. Since ad - bc is the determinant, a determinant of zero would yield a denominator of zero in the inverse formula, and you can't divide by 0.

Hope this helps

Answer 6

so that one doesn't have an inverse?

Answer 7

there's an attachment ^

Answer 8

1*4-(-2)*2 = ?

Answer 9

8

Answer 10

so?

Answer 11

if the det is non zero then the matrix has an inverse

Answer 12

OH! so yes it has an inverse then

Answer 13

so it just depends on the determinant correct?

Answer 14

yes, but that is not always so easy to find, so you might use something else on that list.

Answer 15

but its easy enough for a 2x2 matrix

Answer 16

okay thank you!