Math 225 midterm 2 Exam Questions And
Answers Latest 2025
The Invertible Matrix Theorem -
correct answer ✅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
_________ -
correct answer ✅invertible with B=A⁻¹ and A=B⁻¹
, Math 225 midterm 2 Exam Questions And
Answers Latest 2025
Let T: Rⁿ→Rⁿ be a linear transofrmation and let A be the standard
matrix for T, then -
correct answer ✅T is invertible iff A is an invertible matrix.
a 2x2 matrix is invertible iff its determinant is _____ -
correct answer ✅nonzero
if A is a triangular matrix, then the det A is the product of -
correct answer ✅the entries on the main diagonal of A
Row Operations Theorem: let A be a square matrix -
correct answer ✅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 -
correct answer ✅det A≠0
If A is an nxn matrix, then det A^T= -
correct answer ✅det A
Answers Latest 2025
The Invertible Matrix Theorem -
correct answer ✅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
_________ -
correct answer ✅invertible with B=A⁻¹ and A=B⁻¹
, Math 225 midterm 2 Exam Questions And
Answers Latest 2025
Let T: Rⁿ→Rⁿ be a linear transofrmation and let A be the standard
matrix for T, then -
correct answer ✅T is invertible iff A is an invertible matrix.
a 2x2 matrix is invertible iff its determinant is _____ -
correct answer ✅nonzero
if A is a triangular matrix, then the det A is the product of -
correct answer ✅the entries on the main diagonal of A
Row Operations Theorem: let A be a square matrix -
correct answer ✅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 -
correct answer ✅det A≠0
If A is an nxn matrix, then det A^T= -
correct answer ✅det A
