Linear Algebra Quiz: Verifying Solutions And Identities

What this quiz covers

This quiz focuses on Verifying Solutions And Identities, giving you a quick way to practice the rules, question types, and explanations that matter most for Linear Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Solve the system using Gaussian elimination and compare the result
2. Check if $Ax = b$ by computing the matrix-vector product $Ax$ (correct answer)
3. Verify that $x$ is in the column space of $A$
4. Check if $A^{-1}b = x$ by computing the inverse of $A$

Question 2

1. Computing $(AB)^T$ by transposing $A$ and $B$ first, then multiplying
2. Computing $B^T A^T$ as $A^T B^T$ due to commutativity assumption (correct answer)
3. Making arithmetic errors in the matrix multiplication $AB$
4. Incorrectly transposing one of the matrices $A$ or $B$

Question 3

1. Check if $A$ and $B$ have the same characteristic polynomial
2. Verify that $A$ and $B$ have the same eigenvalues with same multiplicities
3. Find an invertible matrix $P$ such that $B = P^{-1}AP$ (correct answer)
4. Compare all invariants: determinant, trace, and eigenvalues of $A$ and $B$

Question 4

1. Compute $A^2$ and verify that $A^2 = 3A - 2I$
2. Compute $A^2$ and verify that $A^2 - 3A = -2I$ (correct answer)
3. Verify that $\det(A^2 - 3A + 2I) = 0$
4. Check that the eigenvalues of $A$ satisfy $\lambda^2 - 3\lambda + 2 = 0$

Question 5

1. Compute $\text{rank}(A)$, $\text{rank}(B)$, and $\text{rank}(AB)$, then check the inequality (correct answer)
2. Show that $\text{col}(AB) \subseteq \text{col}(A)$ and $\text{row}(AB) \subseteq \text{row}(B)$
3. Verify that $\text{nullity}(AB) \geq \max(\text{nullity}(A), \text{nullity}(B))$
4. Compute determinants and check if $|\det(AB)| \leq \min(|\det(A)|, |\det(B)|)$

Question 6

1. Show that $\det(A - 3I) = 0$ and $A\mathbf{v} = 3\mathbf{v}$
2. Verify only that $A\mathbf{v} = 3\mathbf{v}$ since this confirms both claims
3. Compute the characteristic polynomial and check that $3$ is a root
4. Show that $(A - 3I)\mathbf{v} = \mathbf{0}$ and $\mathbf{v} \neq \mathbf{0}$ (correct answer)

Question 7

1. Verify that $PP^T = I$ to ensure $P$ is square and invertible (correct answer)
2. Check that $\det(P) = \pm 1$ to confirm the orthogonal property
3. Verify that all columns of $P$ have unit length independently
4. Confirm that $P^{-1} = P^T$ by computing the inverse directly

Question 8

1. $AB + AC$ and verify it equals $$ \begin{pmatrix} 3 & 3 \ 9 & 3 \end{pmatrix} $$$$ (correct answer)
2. $AB$ and $AC$ separately, then check if their sum gives the expected result
3. $(B + C)A$ to verify that multiplication is commutative in this case
4. $BA + CA$ and compare with $A(B + C)$ to check both distributive laws

Question 9

1. Verify that the vectors span $\mathbb{R}^3$ by showing any vector can be written as their linear combination
2. Check that each vector has unit length to ensure orthogonality
3. No additional verification needed since three linearly independent vectors in $\mathbb{R}^3$ automatically span $\mathbb{R}^3$ (correct answer)
4. Verify that the vectors are orthogonal to each other

Question 10

1. Check if the coefficient matrix has full rank
2. Check if $\{\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}\}$ are linearly independent
3. Determine if $\mathbf{b}$ is in the span of $\{\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}\}$
4. Verify that $2\mathbf{v_1} + 1\mathbf{v_2} + 1\mathbf{v_3} = \mathbf{b}$ component-wise (correct answer)