Inverse matrices and systems of equations – further maths test to download as PDF

(a) Find the inverse of \(\mathbf A=\begin{pmatrix}2&3\\1&4\end{pmatrix}\). [2]

(b) Hence solve \(2x+3y=5\) and \(x+4y=0\). [2]

\(\mathbf A=\begin{pmatrix}1&0&2\\2&1&0\\0&3&1\end{pmatrix}\).

1. Show that \(\det\mathbf A=13\). [1]
2. Find \(\mathbf A^{-1}\). [3]
3. Use \(\mathbf A^{-1}\) to solve \(\mathbf A\mathbf x=\begin{pmatrix}7\\4\\9\end{pmatrix}\). [1]

The system is \(x+y+z=2\), \(2x+y+3z=5\), \(3x+2y+4z=k\).

1. Show that the system has no unique solution for any value of \(k\). [1]
2. Find the value of \(k\) for which the equations are consistent. [1]
3. For that value of \(k\), find the general solution. [2]

The planes are \(\Pi_1\): \(x+2y-z=1\), \(\Pi_2\): \(2x+4y-2z=5\) and \(\Pi_3\): \(x-y+z=0\).

1. Show that \(\Pi_1\) and \(\Pi_2\) are parallel and not the same plane. [2]
2. Explain why the three planes have no common point, and describe the arrangement. [2]

Three bags of fertiliser A and two bags of fertiliser B cost £34. Two bags of A and five bags of B cost £52.

1. Write this as a matrix equation and find the inverse of the coefficient matrix. [2]
2. Find the price of one bag of each. [1]