By considering the image of $(0, 0)$, find $\mathit{q}$.

Hence find the angle and direction of the rotation represented by $\mathit{R}$.

Write down the matrix $\mathit{S}$.

Find the value of $\text{det\hspace{0.05em}}\mathit{M}$.

Calculate det (BA).

Find Q.

Find the values of $\lambda$ for which the matrix (A − $\lambda$I) is singular.

Let .

Find the value of $q$.

By considering the image of $(1, 0)$ and $(0, 1)$, show that

$\mathit{P}=\left(\begin{array}{cc}\frac{\sqrt{3}}{4} & \frac{1}{4}\\ -\frac{1}{4} & \frac{\sqrt{3}}{4}\end{array}\right)$.

Find the value of $a$ and the value of $b$.

The matrices A, B, X are given by

A = , B = , X = , $\text{where}$ $a$, $b$, $c$, $d\in \mathbb{Q}$.

Given that AX + X = Β, find the exact values of $a$, $b$, $c$ and $d$.

Write down these three equations as a matrix equation.

Three drones are initially positioned at the points $\text{A}$, $\text{B}$ and $\text{C}$. After performing the movements listed above, the drones are positioned at points $\text{A}\prime$, $\text{B}\prime$ and $\text{C}\prime$ respectively.

Show that the area of triangle $\text{ABC}$ is equal to the area of triangle $\text{A}\prime \text{B}\prime \text{C}\prime$ .

Find ${\mathit{P}}^2$.

Find a single transformation that is equivalent to the three transformations represented by matrix $\mathit{P}$.

Write down an equation satisfied by $\left(\begin{array}{c}a\\ b\end{array}\right)$.

Use $\mathit{P}=\mathit{R}\mathit{S}$ to find the matrix $\mathit{R}$.

Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.

Find a single matrix $\mathit{P}$ that defines a transformation that represents the overall change in position.

Hence state what the value of ${\mathit{P}}^2$ indicates for the possible movement of the drone.

Find the matrix X, such that AX = C.

Write down the value of $q$.

(A + B)² = A² + 2AB + B²

Use the result from part (b) (ii) to explain why A is non-singular.

Hence find A⁻¹B.

CA = B  C = $\frac{B}{A}$

The square matrix X is such that X³ = 0. Show that the inverse of the matrix (I – X) is I + X + X².

Show that A⁴ = 12A + 5I.

Find the value of $b$.

Given that M³ can be written as a quadratic expression in M in the form aM² + bM + cI , determine the values of the constants a, b and c.

Find a quadratic expression in M for the inverse matrix M^–1.

Write down A⁻¹.

Find D.

Find T₄.

Find the matrix DA.

Write down the inverse matrix A⁻¹.

Using your results from part (a) (ii), find T₁₀₀.

Write down the value of $b$.

find AB.

Find $S$₄.

Write down the value of $a$.

write down A + B.

Write these equations as a matrix equation.

Express X in terms of A⁻¹ and B.

 Write down M⁻¹.

Write down the determinant of M.

A^–1 = I – A.

Find det A.

 (A – kI)³ = A³ – 3kA² + 3k²A – k³

Show that $27m+9n+3p=18$.

Using this value of $a$, find $M^{-1}$ and hence solve the system of equations:

$-x+2y=-3$

$2x-y=3$

A³ = –I.

Find X.

Show that ${\gamma }^2=\gamma -1$.

Find A + B.

Find M².

Write down $M^{-1}$ for this value of $a$.

Given that XA + B = C, express X in terms of A^–1, B and C.

If A =  and B = , find 2 values of $x$ and $y$, given that AB = BA.

Hence find the value of ${\left(1-\gamma \right)}^6$.

Find $M^2$ in terms of $a$.

Find B if AB = C.

If all of the elements of M are positive, find the range of possible values for $a$.

Show that M⁴ = 19M² + 40M + 30I.

Solve the matrix equation.

Find a relationship between $a$ and $b$ if the matrices  and  commute under matrix multiplication.

It is now given that A is singular.

By considering appropriate determinants, prove that f is not a bijection.

$A$ and $B$ are 2 × 2 matrices, where  and . Find $B$

 Calculate M².

Write down the value of $a$.

Given that A is non-singular, prove that f is a bijection.

does X^–1 + Y^–1 have an inverse? Justify your conclusion.

Hence or otherwise solve

$$x-3y=1$$

$$2x+z=2$$

$$4x+y+3z=-1$$

The function $f$ can also be written $f(x)=x\left(x-1\right)\left(rx-s\right)$ where $r$ and $s$ are integers. Find $r$ and $s$.

Find the value of $a$ if the determinant of matrix $M$ is −1.

find X and Y.

Show that M³ = .

Write down M⁴.

Hence, find the point of intersection of the three planes.

$$\begin{array}{c}x-3y+z=1\\ 2x+2y-z=2\\ x-5y+3z=3\end{array}$$

and hence find an expression for $A^n+{\left(A^n\right)}^{-1}$.

Consider the matrix A , where $x\in \mathbb{R}$.

Find the value of $x$ for which A is singular.

Hence solve M$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}4\\ 8\end{array}\right)$.

Write down the inverse of the matrix A = .

Hence, solve for X.

Show that M is non-singular.

Given that 2A + B = , find the value of $p$ and of $q$.

Use the values from part (b) (i) to express A⁴ in the form $p$A+ $q$I where $p$, $q\in \mathbb{Z}$.

Find the set of values of $k$ for which the system has a unique solution.

Given matrices A, B, C for which AB = C and det A ≠ 0, express B in terms of A and C.

Find the coordinates of the point of intersection of the planes $x+2y+3z=5$, $2x-y+2z=7$, $3x-3y+2z=10$.

Write down the other two linear equations in $m$, $n$ and $p$.

Hence, or otherwise, find A^–1.

Using mathematical induction, prove that Mⁿ can be written as a quadratic expression in M for all positive integers n ≥ 3.

Find the values of $a$ and $b$ given that the matrix  is the inverse of the matrix .

Assuming that $\mathit{A}$ is non-singular, use the result in part (b)(i) to show that

${\mathit{A}}^{-1}=\frac{1}{\beta }\left(\alpha \mathit{I}-\mathit{A}\right)$.

Find the value of $m$ and of $n$.

Given that B , and C , find X.

Find A⁻¹.

The matrix C = $\left(\begin{array}{c}20\\ 28\end{array}\right)$ and 2AB = C. Find the value of $x$.

For the values of $a$ and $b$ found in part (a), solve the system of linear equations

$$\begin{array}{c}x+2y-2z=5\\ 3x+by+z=0\\ -x+y-3z=a-1.\end{array}$$

Find D^–1.

Find $S$₁₀₀.

Write down the inverse of the matrix

A = 

det (2A − B).

Find CD.

The square matrix X is such that X³ = 0. Show that the inverse of the matrix (I – X) is I + X + X².

Find AB.

find $x$ and $y$ in terms of $n$.

2A − B.

Write down the inverse, A^–1.

Verify that

${\mathit{A}}^2-\alpha \mathit{A}+\beta \mathit{I}=$ 0.

Find the values of the real number $k$ for which the determinant of the matrix  is equal to zero.

Hence solve the system of equations

$$x+2y+z=0$$

$$x+y+2z=7$$

$$2x+y+z=17$$

Hence show that M is a singular matrix.

Solve this matrix equation.

Find AB.

A fourth plane with equation $x+y+z=d$ passes through the point of intersection. Find the value of $d$.

Find the inverse of the matrix .

Given that AB = , find $a$.

Find the matrix A².

If $M^2$ is equal to , find the value of $a$.

Given that A =  and B = , find X if BX = A – AB.

Show that (I − M)² = I − M where I is the identity matrix.

Hence show that I = $\frac{1}{5}$A (6I – A).

Given that M =  and that M² – 6M + kI = 0 find k.

Show that $a+d=1$.

Find an expression for $bc$ in terms of $a$.

The matrices A, B, C and X are all non-singular 3 × 3 matrices.

Given that A^–1XB = C, express X in terms of the other matrices.

Find −3A. 

Find A².

Let X be a 2 × 2 matrix such that AX = B. Find X.

By considering the determinant of a relevant matrix, show that the eigenvalues, $\lambda$, of $\mathit{A}$ satisfy the equation

${\lambda }^2-\alpha \lambda +\beta =0$,

where $\alpha$ and $\beta$ are functions of $a, b, c, d$ to be determined.

Let B = .

Given that B² – B – 4I = , find the value of $k$.

If  and det $A=14$, find the possible values of $p$.

 Show that det(M²) is positive.

If A = and A² is a matrix whose entries are all 0, find $k$.

 Find A(A^–1B + 2A^–1)A.

Find BA.

If det A² = 16, determine the possible values of $a$.