Matrices and determinants – university Year 1 maths test to download as PDF

Let \(P=\) \(\begin{pmatrix}2&1\\0&3\end{pmatrix}\) and \(Q=\) \(\begin{pmatrix}1&4\\2&0\end{pmatrix}\).

1. Find \(PQ\) and \(QP\). (2 marks)
2. Find \((PQ)^T\). (1 mark)
3. Find \(\det(PQ)\) and check that it equals \(\det P\det Q\). (1 mark)

Find the determinant of \(\begin{pmatrix}1&2&0\\3&1&4\\0&5&2\end{pmatrix}\) and state whether the matrix is invertible. Show your method.

Let \(A=\) \(\begin{pmatrix}4&3\\3&2\end{pmatrix}\).

1. Find \(A^{-1}\). (2 marks)
2. Use it to solve \(4x+3y=1\) and \(3x+2y=1\). (2 marks)

At a concert, 2 adult tickets and 3 child tickets cost £62, and 3 adult tickets and 2 child tickets cost £68.

1. Write the information as a matrix equation. (1 mark)
2. Use Cramer's rule to find the price of each ticket. (3 marks)

Let \(M=\) \(\begin{pmatrix}1&k\\2&6\end{pmatrix}\).

1. Find the value of \(k\) for which \(M\) is singular. (2 marks)
2. For that \(k\), say whether the system \(x+ky=3\), \(2x+6y=6\) has no solution or infinitely many. Justify. (2 marks)