Question

determine whether each matrix has an inverse.

Answer 1

4 8
-3 -2

Answer 2

For a matrix to have an inverse, you need two conditions.
1. The matrix must be square.
2. The determinant of the matrix must be nonzero.

Answer 3

what do you mean by nonzero..? @mathstudent55

Answer 4

wait, ok. so if it was
6 -8
-3 4
the determinant is 0. so there's no inverse? @mathstudent55

Answer 5

If you evaluate the determinant of a square matrix, and the determinant has a value of zero, then that matrix has no inverse.
If a square matrix has a determinant that has a value not equal to zero, then that matrix has an inverse.

Answer 6

Correct on your second matrix.

Answer 7

aif any of these are true, then the matrix IS invertable


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 8

thank you both!

Answer 9

wlcm