Answers: Matrix multiplication with special matrices
These are the answers to Questions: Matrix multiplication with special matrices.
Please attempt the questions before reading these answers!
Q1
1.1. \[\begin{aligned} 3x + 3y &= 4 \\[0.2em] x - y &= 5\end{aligned}\]
1.2. \[\begin{aligned} 3x + 3y &= 4 \\[0.2em] x - y &= 5\\[0.2em] -4x + 3y &= 0\end{aligned}\]
1.3. \[\begin{aligned} 3x + 3y + 3z &= 4 \\[0.2em] y - z &= 5\\[0.2em] -4x + 3y + z &= 0\end{aligned}\]
1.4. \[\begin{aligned} 3x + 3y + 3z &= 4\end{aligned}\]
1.5. \[\begin{aligned} 3x + 3y + 3z + 3t &= 4 \\[0.2em] x - y\;\;\;\qquad + t &= 5\end{aligned}\]
Q2
2.1. \(\displaystyle\quad \begin{bmatrix}9&-1\\1&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}-4\\1\end{bmatrix}\)
2.2. \(\displaystyle\quad \begin{bmatrix}1&3&1\\-1&-1&9\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}8\\-1\end{bmatrix}\)
2.3. \(\displaystyle\quad \begin{bmatrix}3&1\\1&-1\\8&-9\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}4\\5\\1\end{bmatrix}\)
2.4. \(\displaystyle\quad \begin{bmatrix}3&3&3\\0&0&1\\-4&-4&8\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}4\\1\\1\end{bmatrix}\)
2.5. \(\displaystyle\quad \begin{bmatrix}2&2&2&9\\-1&-1&-1&-1\end{bmatrix}\begin{bmatrix}x\\y\\z\\t\end{bmatrix} = \begin{bmatrix}4\\-10\end{bmatrix}\)
Q3
3.1. \(\quad S0_{3\times 9} = 0_{2\times 9}\)
3.2. \(\quad 0_{9\times 3}T = 0_{9\times 2}\)
3.3. \(\quad SI_3 = S\)
3.4. \(\quad I_2S = S\)
3.5. \(\quad I_2T\) is undefined as \(I_2\) has two columns and \(T\) has three rows.
Q4
For 4.1 to 4.4, pick \(A = B = I_2\), the \(2\times 2\) identity matrix. In this case, \(AB = I_2\) which is both upper triangular and lower triangular, a diagonal matrix, and the \(2\times 2\) identity matrix.
For 4.5, pick \[A = \begin{bmatrix}1&1\\0&1\end{bmatrix}\qquad\textsf{ and }\qquad B = \begin{bmatrix}1&0\\1&1\end{bmatrix}\] where \(A\) is upper triangular and \(B\) is lower triangular. But \[AB = \begin{bmatrix}1&1\\0&1\end{bmatrix}\begin{bmatrix}1&0\\1&1\end{bmatrix} = \begin{bmatrix}2&1\\1&1\end{bmatrix}\] which is neither upper nor lower triangular.
Version history
v1.0: initial version created 05/26 by tdhc.