Matrices questions

Questions to attempt

6 questions on Matrices, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. CP-3.1 · Add, subtract and multiply conformable matrices. Multiply a matrix by a scalar.

   Q1

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   1.

   Let $C=\begin{pmatrix}1&2&-1\\0&3&2\end{pmatrix}$ and $D=\begin{pmatrix}2&1\\-1&4\\3&0\end{pmatrix}$. Calculate $CD$ and state its order.

   (4)

   (Total for Question 1 is 4 marks)

   Reveal answer and method

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2. CP-3.2 · Understand and use zero and identity matrices.

   Q2

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   2.

   Let $A=\begin{pmatrix}1&1\\0&2\end{pmatrix}$. Verify that $(A-I)(A-2I)=O$ and hence state a quadratic matrix equation satisfied by $A$.

   (3)

   (Total for Question 2 is 3 marks)

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3. CP-3.3 · Use matrices to represent linear transformations in 2-D. Successive transformations. Single transformations in 3-D.

   Q3

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   3.

   A reflection in the line $y=x$ is followed by a stretch parallel to the $x$-axis with scale factor $2$. Find the matrix of the combined transformation and the image of $(2,-1)$.

   (4)

   (Total for Question 3 is 4 marks)

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4. CP-3.4 · Find invariant points and lines for a linear transformation.

   Q4

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   4.

   Find the invariant lines through the origin for the transformation with matrix $B=\begin{pmatrix}2&1\\0&1\end{pmatrix}$.

   (4)

   (Total for Question 4 is 4 marks)

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5. CP-3.5 · Calculate determinants of 2x2 and 3x3 matrices and interpret as scale factors, including the effect on orientation.

   Q5

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   5.

   Calculate the determinant of $M=\begin{pmatrix}1&2&0\\-1&3&1\\2&0&1\end{pmatrix}$. Hence state the volume scale factor and the effect on orientation.

   (4)

   (Total for Question 5 is 4 marks)

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6. CP-3.6 · Understand and use singular and non-singular matrices. Properties of inverse matrices. Calculate and use the inverse of non-singular 2x2 and 3x3 matrices.

   Q6

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   6.

   Find the inverse of $C=\begin{pmatrix}1&1&0\\0&1&1\\1&0&1\end{pmatrix}$.

   (5)

   (Total for Question 6 is 5 marks)

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CP-3 Matrices revision notes