DP Mathematics: Applications and Interpretation Questionbank

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.

Verify that

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

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

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

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

Find ${\mathit{P}}^2$.

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

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

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

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

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

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

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

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

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

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 $d$.

find AB.

Hence, solve for X.

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

Find D.

Write down the value of $a$.

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

$-x+2y=-3$

$2x-y=3$

Write these equations as a matrix equation.

Write down the value of $a$.

Show that M³ = .

Let .

Find the value of $q$.

Write down the value of $b$.

Write down the inverse matrix A⁻¹.

Find $S$₁₀₀.

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

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

Given that B , and C , find X.

Find $M^2$ in terms of $a$.

Find CD.

Write down A⁻¹.

Express X in terms of A⁻¹ and B.

Solve the matrix equation.

write down A + B.

Find D^–1.

Write down M⁴.

Find T₄.

Find Q.

Find the value of $b$.

Find $S$₄.

Find M².

Write down the inverse, A^–1.

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

Find X.

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

Solve this matrix equation.

det (2A − B).

Find A + B.

Find AB.

Write down these three equations as a matrix equation.

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

Write down the value of $q$.

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

Find A².

 Write down M⁻¹.

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

Write down the determinant of M.

Find −3A. 

Hence find A⁻¹B.

Hence or otherwise solve

$$x-3y=1$$

$$2x+z=2$$

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

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

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

2A − B.

Find AB.

Find A⁻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 $s$.

Write down the inverse of the matrix A = .

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

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

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

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

A^–1 = I – A.

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

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

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

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

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

A³ = –I.

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

find X and Y.

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

Show that $a+d=1$.

Hence show that M is a singular matrix.

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

(A + B)² = A² + 2AB + B²

Find the matrix A².

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

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

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

 Find A(A^–1B + 2A^–1)A.

 Calculate M².

Hence, or otherwise, find A^–1.

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

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

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

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

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

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

Find the matrix X, such that AX = C.

Find the matrix DA.

Find BA.

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 the determinant of A, where A = .

Write down the inverse of the matrix

A = 

 Show that det(M²) is positive.

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

Calculate det (BA).

Show that M is non-singular.

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

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

 (A – kI)³ = A³ – 3kA² + 3k²A – k³

CA = B  C = $\frac{B}{A}$

Given that AB = , find $a$.

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

Find B if AB = C.

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

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

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

Find the inverse of the matrix .

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

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

Find det A.

Find the value of $m$ and of $n$.

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

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

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

Hence solve the system of equations

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

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

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

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

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

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.

Show that M⁴ = 19M² + 40M + 30I.

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

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

Show that A⁴ = 12A + 5I.

Let B = .

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

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

It is now given that A is singular.

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