Math 225 Exam #2 Questions And Answers
Latest 2025
Invertible Matrix Theorem -
correct answer ✅Let A be a square n x n matrix, then the following
statements are either all true or all false...
-A is an invertible matrix
-A is row equivalent to the n x n identity matrix
-A has n pivot positions
-The equation Ax=0 has only the trivial solution
-The columns of A form a linearly independent set
-The SLE Ax=b has a solution for each b in Rn
-The columns of A span Rn
-There is an n x n matrix C such that CA=I
-There is an n x n matrix D such that AD=I
-A^T is an invertible matrix
Determinant -
correct answer ✅det A= a11detA11 - a12detA12 + ... + (-
1)^(1+n)a1ndetA1n
The determinant of an n x n matrix A can be computed by... -
correct answer ✅... a cofactor expansion across any row or down
any column
, Math 225 Exam #2 Questions And Answers
Latest 2025
Cofactor expansion across the ith row -
correct answer ✅det A= ai1Ci1 + ai2Ci2 + ... + ainCin
=(-1)^(i+1)*ai1*detAi1 + (-1)^(i+2)*ai2*detAi2+ ... + (-
1)^(i+n)*ain*detAin
Cofactor expansion down jth column -
correct answer ✅det A= a1jC1j + a2jC2j + ... + anjCnj
=(-1)^(1+j)*a1j*detA1j + (-1)^(2+j)*a2j*detA2j+ ... + (-
1)^(n+j)*anj*detAnj
If A is a triangular matrix,... -
correct answer ✅... then det A is the product of the entries on the
main diagonal of A
Effects of row operations on determinants -
correct answer ✅Let A be a square matrix
-If a multiple of one row of A is added to another row to produce a
matrix B, then det B= det A
-If two rows are interchanged (swapped) to produce B, then det B=
-det A
Latest 2025
Invertible Matrix Theorem -
correct answer ✅Let A be a square n x n matrix, then the following
statements are either all true or all false...
-A is an invertible matrix
-A is row equivalent to the n x n identity matrix
-A has n pivot positions
-The equation Ax=0 has only the trivial solution
-The columns of A form a linearly independent set
-The SLE Ax=b has a solution for each b in Rn
-The columns of A span Rn
-There is an n x n matrix C such that CA=I
-There is an n x n matrix D such that AD=I
-A^T is an invertible matrix
Determinant -
correct answer ✅det A= a11detA11 - a12detA12 + ... + (-
1)^(1+n)a1ndetA1n
The determinant of an n x n matrix A can be computed by... -
correct answer ✅... a cofactor expansion across any row or down
any column
, Math 225 Exam #2 Questions And Answers
Latest 2025
Cofactor expansion across the ith row -
correct answer ✅det A= ai1Ci1 + ai2Ci2 + ... + ainCin
=(-1)^(i+1)*ai1*detAi1 + (-1)^(i+2)*ai2*detAi2+ ... + (-
1)^(i+n)*ain*detAin
Cofactor expansion down jth column -
correct answer ✅det A= a1jC1j + a2jC2j + ... + anjCnj
=(-1)^(1+j)*a1j*detA1j + (-1)^(2+j)*a2j*detA2j+ ... + (-
1)^(n+j)*anj*detAnj
If A is a triangular matrix,... -
correct answer ✅... then det A is the product of the entries on the
main diagonal of A
Effects of row operations on determinants -
correct answer ✅Let A be a square matrix
-If a multiple of one row of A is added to another row to produce a
matrix B, then det B= det A
-If two rows are interchanged (swapped) to produce B, then det B=
-det A
