Questions: Introduction to matrices

Summary

A selection of questions for the study guide on introduction to matrices.

Before attempting these questions, it is highly recommended that you read Guide: Introduction to matrices.

Q1

You are given the following matrices:

\[A = \begin{bmatrix} 2 & -1 & \sqrt{3} \\ 0 & 4 & -\pi \end{bmatrix} \textsf{ , }\quad B = \begin{bmatrix} 2 & -2 \\ 3 & -4 \end{bmatrix} \textsf{ , }\quad C = \begin{bmatrix} 0 & -1 & 2 & 3\\ 4 & -\sqrt{2} & 1 & -5 \\ 6 & \pi & -7 & 0 \end{bmatrix} \textsf{ , }\quad D = \begin{bmatrix} 3 & -1\\ \sqrt{5} & x\\ y & 1/2 \end{bmatrix},\]

\[E = \begin{bmatrix} 1 & -2 & \sqrt{7}\\ 1 & 3 & -4\\ 5 & -6 & 7 \end{bmatrix} \textsf{ , }\quad F = \begin{bmatrix} -2 & 3/4 & -1\\ \pi & -\sqrt{3} & x^2 \\ 7 & 0 & -5 \end{bmatrix} \textsf{ , }\quad G = \begin{bmatrix} -1 \\ 5 \\ 1 \\ 8 \\ 3 \end{bmatrix} \textsf{ , }\quad H = \begin{bmatrix} \sqrt{2} & -3 & 4\\ 5 & -1 & 2/3\\ x & \pi & -7 \\ 8 & 9 & -10 \end{bmatrix}.\]

1.1. Give the dimensions of all matrices \(A-H\).

1.2. Give the values of the following entries:

1. \(\quad [A]_{11}\)

2. \(\quad [G]_{41}\)

3. \(\quad [D]_{12}\)

4. \(\quad [F]_{32}\)

5. \(\quad [B]_{21}\)

6. \(\quad [A]_{12}\)

7. \(\quad [C]_{23}\)

8. \(\quad [E]_{23}\)

9. \(\quad [H]_{31}\)

10. \(\quad [H]_{13}\)

11. \(\quad [E]_{32}\)

12. \(\quad [G]_{11}\)

1.3. Give the main diagonals of the matrices \(A\), \(C\), \(E\), and \(G\).

Q2

You are given the following matrices:

\[ X = \begin{bmatrix} 3 \\ -\sqrt{5} \\ 2 \\ \pi \end{bmatrix} \textsf{ , }\quad Y = \begin{bmatrix} -1 \\ 5 \\ 2/3 \\ -\sqrt{7} \end{bmatrix} \textsf{ , }\quad Z = \begin{bmatrix} 1 & -2 & 3 \\ 4 & 1 & -5 \\ 6 & -7 & \pi \end{bmatrix} \textsf{ , }\quad W = \begin{bmatrix} \sqrt{3} & -4 & 5/2 \\ -6 & 7 & -8 \\ 9 & -1 & 10 \end{bmatrix},\]

\[M = \begin{bmatrix} 1 & -2 & 3 & 4\\ 5 & -1 & \sqrt{2} & -6 \end{bmatrix} \textsf{ , }\quad N = \begin{bmatrix} -\pi & 3/4 & -1 & 2 \\ 0 & -\sqrt{5} & x & 7 \end{bmatrix} \textsf{ , }\quad O = \begin{bmatrix} 1 & -2 \\ 3 & 4 \\ -5 & 1 \end{bmatrix} \textsf{ , }\quad P = \begin{bmatrix} \sqrt{3} & -4 \\ \pi & 5 \\ -6 & 7 \end{bmatrix}.\]

Calculate the following questions using matrix addition, subtraction, and scalar multiplication:

2.1. \(\quad X+Y\)

2.2. \(\quad Z-W\)

2.3. \(\quad N+M\)

2.4. \(\quad O-P\)

2.5. \(\quad 3X\)

2.6. \(\quad -2Y\)

2.7. \(\quad xZ\)

2.8. \(\quad -4W\)

2.9. \(\quad yM\)

2.10. \(\quad 7N\)

2.11. \(\quad (1/2)O\)

2.12. \(\quad -4P\)

2.13. \(\quad 3X+Y\)

2.14. \(\quad -2(Z+W)\)

2.15. \(\quad N-4M\)

Q3

You are given the following matrices:

\[Q = \begin{bmatrix} 2 & 3 & 1 & 4 \\0 & 3 & 1 & 4\\ 0 & 0 & 1 & 4\\ 0 & 0 & 0 & 4 \end{bmatrix} \textsf{ , }\quad R = \begin{bmatrix} -1 & 1 \\ 3 & -3 \\ \pi & -2\pi \\ 5 & 15\end{bmatrix} \textsf{ , }\quad S = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 0 & 3 \end{bmatrix} \textsf{ , }\quad T = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 8\end{bmatrix},\]

Give their main diagonals, and state whether each of the matrices are:

3.1. square;

3.2. upper triangular;

3.3. lower triangular;

3.4. diagonal.

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After attempting the questions above, please click this link to find the answers.

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Version history

v1.0: initial version created 04/25 by Jessica Taberner as part of a University of St Andrews VIP project.

This work is licensed under CC BY-NC-SA 4.0.