Invertible Matrix Theorem Study guide
verified to pass 2026/2027
Invertible Matrix Theorem - correct answer ✔Let A be an nxn matrix. Then the following statements
are equivalent. That is, for given A, the statements are either all true or all false.
1st Term: a - correct answer ✔A is an invertible matrix
b - correct answer ✔A is row equivalent to the nxn identity Matrix
c - correct answer ✔A has n pivot positions
d - correct answer ✔The equation Ax=0 has only trivial solutions
e - correct answer ✔The columns of A form a linearly independent set
f - correct answer ✔The linear transformation x -->Ax
g - correct answer ✔The equation Ax=b has at least one solution for each b in R^n
h - correct answer ✔The columns of A span R^n
i - correct answer ✔The linear transformation x-->Ax maps R^n onto R^n
j - correct answer ✔There is an nxn matrix C such that CA=I
verified to pass 2026/2027
Invertible Matrix Theorem - correct answer ✔Let A be an nxn matrix. Then the following statements
are equivalent. That is, for given A, the statements are either all true or all false.
1st Term: a - correct answer ✔A is an invertible matrix
b - correct answer ✔A is row equivalent to the nxn identity Matrix
c - correct answer ✔A has n pivot positions
d - correct answer ✔The equation Ax=0 has only trivial solutions
e - correct answer ✔The columns of A form a linearly independent set
f - correct answer ✔The linear transformation x -->Ax
g - correct answer ✔The equation Ax=b has at least one solution for each b in R^n
h - correct answer ✔The columns of A span R^n
i - correct answer ✔The linear transformation x-->Ax maps R^n onto R^n
j - correct answer ✔There is an nxn matrix C such that CA=I
