How do you determine if a matrix is invertible by investigating the equation Ax = I?
For a 3×3 matrix (A) with the following
Row 1: 1, 0, 1
Row 2: 1, 1, 0
Row 3: 0, 1, 1
I know the identity for a 3×3 matrix is
Row 1: 1, 0, 0
Row 2: 0, 1, 0
Row 3: 0, 0, 1
Also, I know the corresponding x must be of the form 3×3 to ensure the multiplication is valid to produce the identity matrix.
Therefore, x will have some form associated with:
Row 1: x11, x12, x13
Row 2: x21, x22, x23
Row 3: x31, x32, x33
Not sure how to proceed from here to find out whether the matrix is invertible.
I know for a 2×2 matrix I can tell whether the matrix is invertible by examining the determinant such that if the determinant is 0 then the matrix is said to be singular, hence has no inverse. Does this property hold for a 3×3 matrix?
Best Answer
You are correct that a real valued square matrix is invertible $\iff$ its determinant is nonzero. Also, $A$ represents a linear transformation between vector spaces of the same dimension, so it is invertible $\iff$ $\ker(A) = \{0\}$. The second method corresponds to row reducing A.