Question Type 7: Representing a system of linear equations using matrix notation Exercises

A chemical mix requires reagents $A,B,C$ in amounts $x,y,z$ grams satisfying: $\begin{array}{rcl} 2x + y - z & = & 4 \\ x + 3y + 2z & = & 7 \\ 4x - y + 5z & = & 12 \end{array}$

Form the augmented matrix for this system.

An investor allocates funds $x,y,z$ (thousands £) into three projects. Returns satisfy:

Express this system in matrix form and define all matrices.

Express the system $\begin{cases} 2x - 3y = 7 \\ 4x + y = 1 \end{cases}$ in matrix form $AX = B$.

A logistics firm packs small ($s$), medium ($m$), and large ($l$) boxes. Total weight and volume constraints give: $\begin{aligned} 5s + 10m + 20l &= 200 \\ 2s + 4m + 5l &= 50 \end{aligned}$ Express these two equations in matrix form $AX = B$, where $X = (s,m,l)^T$.

Given the augmented matrix $\begin{pmatrix}1 & -2 & 3 & 5\\0 & 4 & -1 & 2\\2 & 0 & 1 & 6\end{pmatrix},$ write the corresponding system of equations in $x,y,z$.

Given the matrix equation $\begin{pmatrix}3 & 0 & 1\\1 & 2 & -1\\4 & -1 & 2\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix} = \begin{pmatrix}7\\3\\10\end{pmatrix},$ write out each of the three scalar equations.

Represent the system of equations

in the form $AX = B$, clearly defining each matrix.

Given the system $\begin{aligned} x + 2y - z &= 4 \\ -2x + y + 3z &= 1 \\ 3x - y + 2z &= 5, \end{aligned}$ write it in the form $AX = B$.

Write the system of four equations

in the matrix form $AX = B$.

Let $A = \begin{pmatrix}1 & 2 & 0\\0 & 3 & 4\end{pmatrix},\quad X = \begin{pmatrix}u\\v\\w\end{pmatrix},\quad B = \begin{pmatrix}5\\11\end{pmatrix}.$ Write explicitly the system of linear equations represented by $AX = B$.

A juice requires mixing apple, banana, and cherry concentrates. If $x,y,z$ are volumes (litres) of each concentrate, the recipe demands: $\begin{aligned} x + 2y + z &= 10 \\ 2x - y + 3z &= 15 \\ 3x + y - z &= 5 \end{aligned}$ Express this system in matrix notation.