On a clay tablet from Babylon, approx. 2000 BC, the following problem is stated:
A man who is 30 years older than his son will in 8 years be 4 times as old as the son. How old are the farther and the son?
A
Provide two equations with two unknowns that contain the information from the clay tablet. Reduce the augmented matrix corresponding to the system of equations and find in this way the solution.
B
Check that your solution fits the information on the clay tablet.
Which form should the matrix $\,\mathbf X\,$ have for $\,(*)\,$ to be meaningful? Solve the equation using Gauss-Jordan elimination and compare with the result from question a).
hint
“Solve the equation” means: Find $\,\mathbf X\,$ which must be a $\,2 \times 2\,$ matrix.
C
Justify that $\,\mathbf A\,$ is invertible. Solve $\,(*)\,$ by isolating $\,\mathbf X\,$ on the left-hand-side of $\,(*)\,.$
hint
First you must find $\,\mathbf A^{-1}\,.$ Do this by Gauss-Jordan elimination.
hint
That is, you should reduce the augmented matrix $\,\mT=\left[\mathbf A\,|\,\mathbf E\right]\,$ where $\,\mathbf E\,$ is the identity matrix in $\,\Bbb R^{2 \times 2}\,.$
hint
Now you should have $\,\mathrm{rref}(\mT)=\left[\mathbf E\,|\,\mathbf A^{-1}\right]\,.$
By hand: Show that $ \mA\mB $ is singular (i.e. not regular), an therefore you cannot determine $ (\mA\mB)^{-1} $.
hint
First compute the product $\mA \cdot \mB$.
hint
What does it mean that a matrix is regular? See if necessary the first pages in eNote 8.
hint
Find the rank of the matrix. If the rank is equal to the number of columns, the matrix is regular.
hint
Is $\mA\mB$ square?
hint
Now that the matrix $\mA\mB$ is square, the inverse matrix can be computed, see Method 8.4. But there are further requirements that must be fulfilled.
hint
The matrix must be both square and regular. Was it regular?
answer
The rank of $\mA\mB$ is 2, i.e. less than the number of columns. So, the matrix is not regular, but singular, and therefore the inverse matrix cannot be found.
Exercise 5: The Structure of Solution Sets
Let us consider the system of equations \begin{equation}
\begin{aligned}
x_1 + x_2 + 2x_3 &= 3\newline
2x_1 - x_2 + 4x_3 &= 0\newline
x_1 + 3x_2 - 2x_3 &= 3\newline
-3x_1 - 2x_2 + x_3 &= 0\newline
\end{aligned}
\end{equation}
A
Find the reduced row echelon form of the augmented matrix corresponding to the given system of equations. From this determine the rank of both the coefficient matrix and the augmented matrix. How many solutions does the system have? Compute the solutions.
answer
$(x_1, x_2, x_3)=(-1, 2, 1)$.
B
State the general solution to the homogeneous linear system of equations that correspond to the given inhomogeneous system of equations.
Now we will consider the system of equations \begin{equation}
\begin{aligned}
x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 &= 1\newline
2x_1 + 3x_2 + 4x_3 + 5x_4 + x_5 &= 2\newline
3x_1 + 4x_2 + 5x_3 + 6x_4 - 3x_5 &= 3\newline
\end{aligned}
\end{equation}
C
Find the reduced row echelon form of the augmented matrix corresponding to the new system of equations. From this compute the rank of both the coefficient matrix and the augmented matrix. How many solutions does the system have? State these solutions on standard parametric form.
hint
What is the highest possible rank of the augmented matrix? And how many free parameters are there?