CP-3 Matrices — 9FM0 spec coverage

CP-3.1

Add, subtract and multiply conformable matrices. Multiply a matrix by a scalar.

Matrices can be added or subtracted only when they have the same order; combine corresponding entries.
The product $AB$ exists when the number of columns of $A$ equals the number of rows of $B$ ; each entry is a row-column scalar product.
For example, a $2\times3$ matrix multiplied by a $3\times2$ matrix gives a $2\times2$ matrix, while reversing the order gives a $3\times3$ matrix.
A common error is to multiply corresponding entries or assume $AB=BA$ ; matrix multiplication is generally not commutative.

Tier 1 · Easy

2 marks

ORIGINAL

Given $A=\begin{pmatrix}2&-1\\3&0\end{pmatrix}$ and $B=\begin{pmatrix}-1&4\\2&5\end{pmatrix}$ , calculate $2A-B$ .

Tier 2 · Standard

4 marks

ORIGINAL

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.

Tier 3 · Hard

4 marks

ORIGINAL

Given $A=\begin{pmatrix}1&-2\\3&0\end{pmatrix}$ and $B=\begin{pmatrix}p&1\\-1&q\end{pmatrix}$ , the matrix $AB+2A$ is $\begin{pmatrix}8&-9\\18&3\end{pmatrix}$ . Determine $p$ and $q$ .

CP-3.2

Understand and use zero and identity matrices.

The zero matrix $O$ has every entry equal to zero and is the additive identity: $A+O=A$ for matrices of the same order.
The identity matrix $I$ is square, has ones on its leading diagonal and zeros elsewhere, and satisfies $AI=IA=A$ .
Matrix polynomial identities are simplified just like algebraic ones, but each scalar constant is represented by a scalar multiple of $I$ .
A common error is to replace $I$ by the scalar $1$ or to use an identity matrix of the wrong order.

Tier 1 · Easy

2 marks

ORIGINAL

For $A=\begin{pmatrix}2&-1\\0&3\end{pmatrix}$ , write down $A+O$ and $IA$ , where $O$ and $I$ have the appropriate order.

Tier 2 · Standard

3 marks

ORIGINAL

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 $A$ .

Tier 3 · Hard

4 marks

ORIGINAL

A square matrix $M$ satisfies $M^2=3M+2I$ . Without finding the entries of $M$ , simplify $M^3-4M^2+M$ .

CP-3.3

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

For a linear transformation, the columns of its matrix are the images of the standard basis vectors.
If transformation $A$ is followed by transformation $B$ , the combined matrix is $BA$ , because $\mathbf{x}\mapsto A\mathbf{x}\mapsto BA\mathbf{x}$ .
In three dimensions, use a $3\times3$ matrix acting on $\begin{pmatrix}x\\y\\z\end{pmatrix}$ ; the specification confines 3-D transformations to reflection in one of the planes $x=0$ , $y=0$ , $z=0$ or rotation about one of the coordinate axes — for example, negating $z$ reflects points in the plane $z=0$ .
A common error is to reverse the order of multiplication for successive transformations.

Tier 1 · Easy

2 marks

ORIGINAL

The matrix $R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ acts on the point $P(3,-2)$ . Find the image of $P$ and describe the transformation.

Tier 2 · Standard

4 marks

ORIGINAL

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 $(2,-1)$ .

Tier 3 · Hard

4 marks

ORIGINAL

A transformation in three dimensions maps $(x,y,z)$ to $(x,-z,y)$ . Write down its matrix, find the image of $(2,-1,4)$ , and explain geometrically what the transformation does.

CP-3.4

Find invariant points and lines for a linear transformation.

An invariant point with position vector $\mathbf{x}$ satisfies $A\mathbf{x}=\mathbf{x}$ , so solve $(A-I)\mathbf{x}=\mathbf{0}$ .
An invariant line is mapped onto itself as a set; its individual points need not all remain fixed.
For a line $y=mx+c$ , transform a general point $(x,mx+c)$ and require the image $(X,Y)$ to satisfy $Y=mX+c$ for every $x$ .
A common error is to test only one point on a proposed invariant line or to confuse an invariant line with a line of invariant points.

Tier 1 · Easy

3 marks

ORIGINAL

Find all invariant points of the transformation with matrix $A=\begin{pmatrix}1&0\\0&3\end{pmatrix}$ .

Tier 2 · Standard

4 marks

ORIGINAL

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

Tier 3 · Hard

6 marks

ORIGINAL

The transformation $T$ maps $(x,y)$ to $(x+y,2y)$ . Find all invariant straight lines and identify any line consisting entirely of invariant points.

CP-3.5

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

For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ , $\det A=ad-bc$ ; a $3\times3$ determinant can be found by cofactor expansion or row reduction.
In two dimensions, areas are multiplied by $|\det A|$ ; in three dimensions, volumes are multiplied by $|\det A|$ .
The sign is geometric: a negative determinant reverses orientation, while a positive determinant preserves it.
A common error is to use the signed determinant as a negative area or volume instead of taking its absolute value for size.

Tier 1 · Easy

3 marks

ORIGINAL

A planar transformation has matrix $A=\begin{pmatrix}1&4\\2&3\end{pmatrix}$ . A region has area $12$ before transformation. Find its image area and state what happens to orientation.

Tier 2 · Standard

4 marks

ORIGINAL

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.

Tier 3 · Hard

5 marks

ORIGINAL

A transformation has matrix $P=\begin{pmatrix}k&2\\3&k-1\end{pmatrix}$ . A triangle of area $7$ is mapped to a triangle of area $42$ . Determine all possible values of $k$ and state which values reverse orientation.

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.

A square matrix is singular when its determinant is zero; otherwise it is non-singular and has a unique inverse.
For a non-singular $2\times2$ matrix, use the determinant formula; for a $3\times3$ matrix, row-reduce $(A\mid I)$ to $(I\mid A^{-1})$ or use $A^{-1}=\frac1{\det A}\operatorname{adj}A$ .
Inverse products reverse order: $(AB)^{-1}=B^{-1}A^{-1}$ , and $AA^{-1}=A^{-1}A=I$ .
A common error is to divide every entry by the determinant without interchanging the diagonal entries and changing the signs of the off-diagonal entries.

Tier 1 · Easy

3 marks

ORIGINAL

Show that $A=\begin{pmatrix}3&1\\2&1\end{pmatrix}$ is non-singular and find $A^{-1}$ .

Tier 2 · Standard

5 marks

ORIGINAL

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

Tier 3 · Hard

4 marks

ORIGINAL

The matrix $D=\begin{pmatrix}2&1\\3&2\end{pmatrix}$ satisfies $D^2-4D+I=O$ . Use this identity, rather than the $2\times2$ inverse formula, to find $D^{-1}$ .

CP-3.7

Solve three linear simultaneous equations in three variables by use of the inverse matrix.

Write three simultaneous equations as $A\mathbf{x}=\mathbf{b}$ , keeping coefficients, variables and constants in a consistent order.
When $A$ is non-singular, left-multiply by $A^{-1}$ to obtain $\mathbf{x}=A^{-1}\mathbf{b}$ .
For example, the rows of $A$ contain the coefficients of $x,y,z$ , while $\mathbf{b}$ contains the three right-hand sides.
A common error is to write $\mathbf{x}=\mathbf{b}A^{-1}$ ; column vectors require multiplication by $A^{-1}$ on the left.

Tier 1 · Easy

3 marks

ORIGINAL

A system is written as $A\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}4\\2\\6\end{pmatrix}$ , where $A^{-1}=\frac12\begin{pmatrix}1&-1&1\\1&1&-1\\-1&1&1\end{pmatrix}$ . Find $x,y,z$ .

Tier 2 · Standard

5 marks

ORIGINAL

Use an inverse matrix to solve $x+2y=5$ , $y+z=4$ and $z=3$ .

Tier 3 · Hard

6 marks

ORIGINAL

The equations $x+y=p$ , $y+z=5$ and $2z=8$ have a solution satisfying $x+y=z$ . Use an inverse matrix to determine $p$ and the solution.

CP-3.8

Interpret geometrically the solution and failure of solution of three simultaneous linear equations.

Each linear equation in $x,y,z$ represents a plane in three-dimensional space.
A unique solution is one point common to all three planes; infinitely many solutions arise when the planes share a common line (a sheaf) or are coincident.
No solution means there is no point common to all three planes — the planes form a prism or are otherwise inconsistent, often because one equation contradicts a linear combination of the others.
A common error is to say that a singular coefficient matrix always means no solution; it may instead give infinitely many solutions.

Tier 1 · Easy

2 marks

ORIGINAL

Interpret geometrically the solution of the three equations $x=1$ , $y=2$ and $z=-3$ .

Tier 2 · Standard

4 marks

ORIGINAL

Consider $x+y+z=1$ , $2x+2y+2z=k$ and $x-y+z=3$ . Describe the solution set geometrically when $k=2$ and when $k\ne2$ .

Tier 3 · Hard

5 marks

ORIGINAL

The planes $x+y+z=2$ , $2x-y+z=1$ and $3x+2z=k$ are given. Determine the value of $k$ for which the three planes contain a common line. State what happens for every other value of $k$ .