Linear Algebra

Single correct

• A512
• B1025
• C1023
• D255

Single correct

• A$n$
• B$n - 1$
• C1
• D$n + 1$

One or more correct

• ADeterminant of $M$ is 1
• BDeterminant of $M$ is 0
• CRank of $M$ is 1
• DThere are at least two non-zero vectors in the null space of $M$

Numerical answer

Numerical answer

One or more correct

• AThere is exactly one value of $k$ for which the above system of equations has no solution.
• BThere exist an infinite number of values of $k$ for which the system of equations has no solution.
• CThere exists exactly one value of $k$ for which the system of equations has exactly one solution.
• DThere exists exactly one value of $k$ for which the system of equations has an infinite number of solutions.

Single correct

• A$1, -1$
• B$2\sqrt{2}, -2\sqrt{2}$
• C$4\sqrt{2}, -4\sqrt{2}$
• D$64\sqrt{2}, -64\sqrt{2}$

Single correct

• A$\begin{pmatrix} 25 & 0 \\ 0 & 25 \end{pmatrix}$
• B$\begin{pmatrix} 125 & 0 \\ 0 & 125 \end{pmatrix}$
• C$\begin{pmatrix} 625 & 0 \\ 0 & 625 \end{pmatrix}$
• D$\begin{pmatrix} 3125 & 0 \\ 0 & 3125 \end{pmatrix}$

Single correct

• A0
• B1
• C2
• D3

One or more correct

• AThe system $PX = Q$ can be solved by first solving $LY = Q$ and then $UX = Y$.
• BIf $P$ is invertible, then both $L$ and $U$ are invertible.
• CIf $P$ is singular, then at least one of the diagonal elements of $U$ is zero.
• DIf $P$ is symmetric, then both $L$ and $U$ are symmetric.

Single correct

• A$-1$
• B0
• C1
• D2

One or more correct

• AThere exist at least $m - n$ linearly independent solutions to this system
• BThere exist $m - n$ linearly independent vectors such that every solution is a linear combination of these vectors
• CThere exists a non-zero solution in which at least $m - n$ variables are 0
• DThere exists a solution in which at least $n$ variables are non-zero

Single correct

• A$G$ is a cycle
• B$G$ is a perfect matching
• C$G$ is a complete graph
• DThere is no such graph $G$

One or more correct

• AThe determinant of $B$ is the negative of the determinant of $A$
• BIf $A$ is invertible, then $B$ is also invertible
• CIf $A$ is symmetric, then $B$ is also symmetric
• DIf the trace of $A$ is zero, then the trace of $B$ is also zero

Single correct

• A$\det(A) = \det(B)$
• B$\det(B) = -\det(A)$
• C$\det(A) = 0$
• D$\det(AB) = \det(A) + \det(B)$

Answer key coming soon

Numerical answer

Answer key coming soon

Single correct

• AStatement 1 is correct and Statement 2 is wrong.
• BStatement 1 is wrong and Statement 2 is correct.
• CBoth Statement 1 and Statement 2 are correct.
• DBoth Statement 1 and Statement 2 are wrong.

Answer key coming soon

Single correct

• A$A^3$
• B$A^3$ divided by 2
• C$A^3$ divided by 3
• D$A^3$ divided by 6

Answer key coming soon

Single correct

• A$L_{32} = 2,\ U_{33} = -\frac{1}{2},\ x_1 = -1$
• B$L_{32} = 2,\ U_{33} = 2,\ x_1 = -1$
• C$L_{32} = -\frac{1}{2},\ U_{33} = 2,\ x_1 = 0$
• D$L_{32} = -\frac{1}{2},\ U_{33} = -\frac{1}{2},\ x_1 = 0$

Answer key coming soon

One or more correct

• AThe diagonal entries of $A^2$ are the degrees of the vertices of the graph.
• BIf the graph is connected, then none of the entries of $A^{n-1} + I_n$ can be zero.
• CIf the sum of all the elements of $A$ is at most $2(n-1)$, then the graph must be acyclic.
• DIf there is at least a 1 in each of $A$'s rows and columns, then the graph must be connected.

Answer key coming soon