Linear Algebra: 2.3 Invertible Matrices
A =nxn is invertible if and only if...
A is row equivalent to I_n Any sequence of elementary row operations reducing A ~ I_n also transforms I_n ~ A^(-1)
If A is an invertible nxn matrix, then for each b in R^n...
Ax = b has the unique solution x = A^(-1)b
An nxn matrix A is said to be invertible if there is an nxn matrix C such that...
CA = I and AC = I
If C is 6x6 and the equation Cx=v is consistent for every v in R^6, is it possible that for some v, the equation Cx=v has more than one solution? Why or why not?
It is not possible. According to the Invertible Matrix Theorem that makes the matrix invertible. Since it is invertible, it has a unique solution.
The algorithm to find A^(-1) for a square matrix
Reduce the system [A I] ~ [I A^(-1)]
If an nxn matrix K cannot be row reduced to I_n, what can you say about the columns of K? Why?
The columns of the matrix are linearly dependent and the columns do not span R^n.
Elementary matrix
a matrix obtained by performing a single elementary row operation on an identity matrix
The determinant of A is written as...
det A = ad - bc
The product of nxn invertible matrices is...
invertible
If ad - bc = 0 then A is...
not invertible
The inverse of AB is the product of the inverses of A and B in ________ order.
reverse
The inverse is _______ and denoted ______.
unique, A^(-1)
