Training Linear Algebra Placement Test Practice — Linear Algebra
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Placement Test Practice — Linear Algebra

24 min Linear Algebra

Placement Test Practice — Linear Algebra

This placement test covers the core linear algebra topics from this module: matrix operations, systems of equations and row reduction, determinants and inverses, and vectors with dot products. It is designed to simulate the difficulty and style of a college-level placement exam or course final. Working through these problems will help you gauge your readiness and identify areas that need more practice.

Try each question from memory first, then verify your answers using the answer key. If you find yourself struggling with a particular topic, revisit the corresponding lesson and its interactive tool for additional practice before retaking the test.

Practice Test — 18 Questions

1. Add two $2 \times 2$ matrices entry-wise.
2. State the size of a $4 \times 1$ matrix.
3. Solve $x+y=4$, $x-y=0$.
4. Interpret a row $[0\;0\;|\;6]$.
5. Find $$\det\begin{bmatrix}1&2\3&5\end{bmatrix}.$$
6. Is a matrix with determinant $0$ invertible?
7. Find the magnitude of $\langle 5,12 \rangle$.
8. Compute $\langle 1,0 \rangle\cdot\langle 0,7 \rangle$.
9. What does dot product $0$ tell you?
10. Multiply $$\begin{bmatrix}1&1\0&2\end{bmatrix}\begin{bmatrix}3\4\end{bmatrix}.$$
11. What is the determinant of $$\begin{bmatrix}2&0\0&2\end{bmatrix}$$?
12. What is the determinant of the identity matrix?
13. Add $\langle 2,3 \rangle + \langle -1,8 \rangle$.
14. Solve $2x+y=8$, $x-y=1$.
15. State one elementary row operation.
16. When is matrix multiplication defined?
17. Why are inverses useful?
18. What object stores the coefficients of a system compactly?
Show Answer Key

1. Matching entries are added.

2. 4 rows, 1 column

3. $(2,2)$

4. No solution

5. $-1$

6. No

7. $13$

8. $0$

9. Perpendicular vectors

10. $$\begin{bmatrix}7\8\end{bmatrix}$$

11. $4$

12. $1$

13. $\langle 1,11 \rangle$

14. $(3,2)$

15. Swap rows; scale a row; add a multiple of one row to another

16. Inner dimensions must match

17. They solve systems and undo linear transformations

18. A coefficient or augmented matrix

Vectors and Dot Products Back to Module