Questions: Matrix multiplication with special matrices
Before attempting these questions, it is highly recommended that you read Guide: Matrix multiplication with special matrices.
Q1
You are given the following matrix multiplications. Express each of these as a system of simultaneous equations.
1.1. \(\displaystyle\quad \begin{bmatrix}3&3\\1&-1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}4\\5\end{bmatrix}\)
1.2. \(\displaystyle\quad \begin{bmatrix}3&3\\1&-1\\-4&3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}4\\5\\0\end{bmatrix}\)
1.3. \(\displaystyle\quad \begin{bmatrix}3&3&3\\0&1&-1\\-4&3&-1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}4\\5\\0\end{bmatrix}\)
1.4. \(\displaystyle\quad \begin{bmatrix}3&3&3\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}4\end{bmatrix}\)
1.5. \(\displaystyle\quad \begin{bmatrix}3&3&3&3\\1&-1&0&1\end{bmatrix}\begin{bmatrix}x\\y\\z\\t\end{bmatrix} = \begin{bmatrix}4\\5\end{bmatrix}\)
Q2
You are given the following systems of simultaneous equations. Express each of these as a matrix multiplication.
2.1. \[\begin{aligned} 9x - y &= -4 \\[0.2em] x + y &= 1\end{aligned}\]
2.2. \[\begin{aligned} x + 3y + z &= 8 \\[0.2em] -x - y + 9z &= -1 \end{aligned}\]
2.3. \[\begin{aligned} 3x + y &= 4 \\[0.2em] x - y &= 5\\[0.2em] 8x - 9y &= 1\end{aligned}\]
2.4. \[\begin{aligned} 3x + 3y + 3z &= 4\\ z &= 1\\-4x - 4y + 8z &= 1\end{aligned}\]
2.5. \[\begin{aligned} 2x + 2y + 2z + 9t &= 4 \\[0.2em] -x - y - z - t &= -10\end{aligned}\]
Q3
Let \(\;\;S = \begin{bmatrix} 1 & -2 & 5 \\ -3 & 4 & -1 \end{bmatrix},\) \(\;\;T = \begin{bmatrix} 5 & -6 \\ 7 & 2 \\ 0 & 8 \end{bmatrix}\). Work out the following matrix products.
3.1. \(\quad S0_{3\times 9}\)
3.2. \(\quad 0_{9\times 3}T\)
3.3. \(\quad SI_3\)
3.4. \(\quad I_2S\)
3.5. \(\quad I_2T\)
Q4
For each of the following statements, give an example of an upper triangular \(2\times 2\) matrix \(A\) and a lower triangular \(2\times 2\) matrix \(B\) such that:
4.1. \(\quad AB\) is upper triangular.
4.2. \(\quad AB\) is lower triangular.
4.3. \(\quad AB\) is diagonal.
4.4. \(\quad AB\) is the \(2\times 2\) identity matrix.
4.5. \(\quad AB\) is neither upper nor lower triangular.
After attempting the questions above, please click this link to find the answers.
Version history
v1.0: initial version created 05/26 by tdhc.