math 225 midterm 2 Questions with Verified
Correct Answers
The Invertible Matrix Theorem
Let A be a square nxn matrix. Then the following statements are equivalent. All are true or all
are false:
a.) A is an invertible matrix
b.) A is row equivalent to the nxn identity matrix
c.) A has n pivot positions
d.) the equation Ax=0 has only the trivial solution
e.) the columns of A form a linearly independent set
f.) the linear transformation x→Ax is one-to-one
g.) the equation Ax=b has at least one solution for each b in Rⁿ
h.) the columns of A span Rⁿ
i.) the linear transformation x→Ax maps Rⁿ onto Rⁿ
j.) there is an nxn matrix C such that CA=I
k.) there is an nxn matrix D such that AD=I
l.) A^T is an invertible matrix
Let A and B be square matrices. If AB=I, then A and B are both _________
invertible with B=A⁻¹ and A=B⁻¹
Let T: Rⁿ→Rⁿ be a linear transofrmation and let A be the standard matrix for T, then
T is invertible iff A is an invertible matrix.
a 2x2 matrix is invertible iff its determinant is _____
nonzero
, if A is a triangular matrix, then the det A is the product of
the entries on the main diagonal of A
Row Operations Theorem: let A be a square matrix
a.) if a multiple of one row of A is added to another row to produce a matrix B then
detB=detA
b.) if two rows of A are interchanged to produce B, then detB=-detA
c.) if one row of A is multiplied by k to produce B, then detB=kdetA
A square matrix A is invertible iff
det A≠0
If A is an nxn matrix, then det A^T=
det A
Multiplicative Property
If A and B are nxn matrices, then detAB=(detA)(detB)
Cramer's Rule: Let A be an invertible nxn matrix.
For any b in Rⁿ, the unique solution x of Ax=b has entries given by xi=detAi(b)/detA
i=1,2,....n
Inverse Formula (Adjugate Formula): Let A be an invertible nxn matrix
A⁻¹=(1/detA)adj.A
If A is a 2x2 matrix, the area of the parallelogram determined by the columns of A is
____.
|detA|
Correct Answers
The Invertible Matrix Theorem
Let A be a square nxn matrix. Then the following statements are equivalent. All are true or all
are false:
a.) A is an invertible matrix
b.) A is row equivalent to the nxn identity matrix
c.) A has n pivot positions
d.) the equation Ax=0 has only the trivial solution
e.) the columns of A form a linearly independent set
f.) the linear transformation x→Ax is one-to-one
g.) the equation Ax=b has at least one solution for each b in Rⁿ
h.) the columns of A span Rⁿ
i.) the linear transformation x→Ax maps Rⁿ onto Rⁿ
j.) there is an nxn matrix C such that CA=I
k.) there is an nxn matrix D such that AD=I
l.) A^T is an invertible matrix
Let A and B be square matrices. If AB=I, then A and B are both _________
invertible with B=A⁻¹ and A=B⁻¹
Let T: Rⁿ→Rⁿ be a linear transofrmation and let A be the standard matrix for T, then
T is invertible iff A is an invertible matrix.
a 2x2 matrix is invertible iff its determinant is _____
nonzero
, if A is a triangular matrix, then the det A is the product of
the entries on the main diagonal of A
Row Operations Theorem: let A be a square matrix
a.) if a multiple of one row of A is added to another row to produce a matrix B then
detB=detA
b.) if two rows of A are interchanged to produce B, then detB=-detA
c.) if one row of A is multiplied by k to produce B, then detB=kdetA
A square matrix A is invertible iff
det A≠0
If A is an nxn matrix, then det A^T=
det A
Multiplicative Property
If A and B are nxn matrices, then detAB=(detA)(detB)
Cramer's Rule: Let A be an invertible nxn matrix.
For any b in Rⁿ, the unique solution x of Ax=b has entries given by xi=detAi(b)/detA
i=1,2,....n
Inverse Formula (Adjugate Formula): Let A be an invertible nxn matrix
A⁻¹=(1/detA)adj.A
If A is a 2x2 matrix, the area of the parallelogram determined by the columns of A is
____.
|detA|
