Answers: Introduction to matrices
These are the answers to Questions: Introduction to matrices.
Please attempt the questions before reading these answers!
Q1
1.1.
The matrix \(A\) has dimension \(2 \times 3\).
The matrix \(B\) has dimension \(2 \times 2\).
The matrix \(C\) has dimension \(3 \times 4\).
The matrix \(D\) has dimension \(3 \times 2\).
The matrix \(E\) has dimension \(3 \times 3\).
The matrix \(F\) has dimension \(3 \times 3\).
The matrix \(G\) has dimension \(5 \times 1\).
The matrix \(H\) has dimension \(4 \times 3\).
1.2.
\(\quad [A]_{11} = 2\)
\(\quad [G]_{41} = 8\)
\(\quad [D]_{12} = -1\)
\(\quad [F]_{32} = 0\)
\(\quad [B]_{21} = 3\)
\(\quad [A]_{12} = -1\)
\(\quad [C]_{23} = 1\)
\(\quad [E]_{23} = -4\)
\(\quad [H]_{31} = x\)
\(\quad [H]_{13} = 4\)
\(\quad [E]_{32} = -6\)
\(\quad [G]_{11} = -1\)
1.3.
\(\quad \textsf{diag}(A) = (2,4)\)
\(\quad \textsf{diag}(C) = (0, -\sqrt{2},-7)\)
\(\quad \textsf{diag}(E) = (1,3,7)\)
\(\quad \textsf{diag}(G) = (-1)\)
Q2
2.1. \(\quad X+Y = \begin{bmatrix} 2 \\ 5-\sqrt{5} \\ 8/3 \\ \pi - \sqrt{7} \end{bmatrix}\)
2.2. \(\quad Z-W = \begin{bmatrix} -1 +\sqrt{3} & -2 & -1/2 \\ -10 & 6 & -3 \\ 3 & 6 & 10 - \pi \end{bmatrix}\)
2.3. \(\quad N+M = \begin{bmatrix} -\pi + 1 & -5/4 & 2 & 6 \\ 5 & 1-\sqrt{5} & x + \sqrt{2} & 1 \end{bmatrix}\)
2.4. \(\quad O-P = \begin{bmatrix} 1 - \sqrt{3} & 2 \\ 3 - \pi & -1 \\ 1 & -6 \end{bmatrix}\)
2.5. \(\quad 3X = \begin{bmatrix} 9 \\ -3\sqrt{5} \\ 6 \\ 3\pi \end{bmatrix}\)
2.6. \(\quad -2Y = \begin{bmatrix} 2 \\-10 \\-4/3 \\2\sqrt{7} \end{bmatrix}\)
2.7. \(\quad xZ = \begin{bmatrix} x & -2x & 3x \\4x & x & -5x \\6x & -7x & \pi x \end{bmatrix}\)
2.8. \(\quad -4W =\begin{bmatrix} \sqrt{3} & -4 & 5/2 \\24 & -28 & 32 \\-36 & 4 & -40 \end{bmatrix}\)
2.9. \(\quad yM = \begin{bmatrix} y & -2y & 3y & 4y\\5y & -y & \sqrt{2}y & -6y\end{bmatrix}\)
2.10. \(\quad 7N =\begin{bmatrix} -7\pi & 21/4 & -7 & 14 \\0 & -7\sqrt{5} & 7x & 49 \end{bmatrix}\)
2.11. \(\quad (1/2)O = \begin{bmatrix} 1/2 & -1 \\3/2 & 2 \\-5/2 & 1/2 \end{bmatrix}\)
2.12. \(\quad -4P =\begin{bmatrix} -4\sqrt{3} & 16 \\-4\pi & -20 \\24 & 28 \end{bmatrix}\)
2.13. \(\quad 3X+Y =\begin{bmatrix} 8 \\ 5 -3\sqrt{5} \\ 20/3 \\ 3\pi - \sqrt{7} \end{bmatrix}\)
2.14. \(\quad -2(Z+W) = \begin{bmatrix} -2 -2\sqrt{3} & 12 & -11 \\4 & -16 & 26 \\-30 & 14 =2 & -20-2\pi \end{bmatrix}\)
2.15. \(\quad N-4M = \begin{bmatrix} -4 -\pi & 35/4 & -13 & -14 \\-20 & 4-\sqrt{5} & x - 4\sqrt{2} & 31 \end{bmatrix}\)
Q3
3.1. \(Q, S, T\) are all square, but \(R\) isn’t.
3.2. \(Q,T\) are upper triangular, but \(R,S\) aren’t.
3.3. \(S,T\) are lower triangular, but \(Q,R\) aren’t.
3.4. \(T\) is diagonal, but \(Q,R,S\) aren’t.
Version history
v1.0: initial version created 04/25 by Jessica Taberner as part of a University of St Andrews VIP project.