math 225 midterm 2 Exam with correct answers

math 225 midterm 2 Exam with correct
answers
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



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



Multiplicative Property - correct answer -If A and B are nxn matrices, then detAB=(detA)(detB)



Cramer's Rule: Let A be an invertible nxn matrix. - correct answer -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 - correct answer
-A⁻¹=(1/detA)adj.A



If A is a 2x2 matrix, the area of the parallelogram determined by the columns of A is ____. - correct
answer -|detA|



If A is a 3x3 matrix, the volume of the parallelepiped determined by the colums of A is____. - correct
answer -|detA|



Let a₁ and a₂ be nonzero vectors. then for any scalar c, the area of the parallelogram determined by a₁
and a₂ equal - correct answer -the area of the parallelogram determine by a₁ and a₂+ca₁



Let T: R²→R² be the linear transformation determined by a 2x2 matrix A. If S is a parallelogram in R²,
then - correct answer -{area of T(S)}=|det A|{area of S}



Let T: R²→R² be the linear transformation determined by a 2x2 matrix A. If T is determined by a 3x3
matrix A, and if S is a parellelepiped in R³, then - correct answer -{volume of T(S)}=|det A|{volume of S}