3x3 Matrices

Solve the simultaneous equations \begin{align} 4x - 2y + 3z &= 8,
2x - 3y + 8z &= -1,
2x + 4y - z &= 0. \end{align} [5]

3 The points \(A ( 1,3 ) , B ( 4,36 )\) and \(C ( 9,151 )\) lie on the curve with equation \(y = p + q x + r x ^ { 2 }\).

1. Using this information, write down three simultaneous equations in \(p , q\) and \(r\).
2. Re-write this system of equations in the matrix form \(\mathbf { C x } = \mathbf { a }\), where \(\mathbf { C }\) is a \(3 \times 3\) matrix, \(\mathbf { x }\) is an unknown vector, and \(\mathbf { a }\) is a fixed vector.
3. By finding \(\mathbf { C } ^ { - 1 }\), determine the values of \(p , q\) and \(r\).

The matrix M is defined as $$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ -1 & -1 & -2 \\ 1 & 2 & c \end{pmatrix}$$ where \(c\) is a real number.

1. The linear transformation T is represented by the matrix \(\mathbf{M}\) Show that, for one particular value of \(c\), the image under T of every point lies in the plane $$x + 5y + 3z = 0$$ State the value of \(c\) for which this occurs. [3 marks]
2. It is given that M is a non-singular matrix.
   1. State any restrictions on the value of \(c\) [2 marks]
   2. Find \(\mathbf{M}^{-1}\) in terms of \(c\) [4 marks]
   3. Using your answer from part (b)(ii), solve $$2x - y + z = -3$$ $$-x - y - 2z = -6$$ $$x + 2y + 4z = 13$$ [3 marks]

6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r }

The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 1 & 5 & 1 \\ 1 & -2 & -2 \\ 2 & 3 & \theta \end{pmatrix}.$$

1. Find the rank of \(\mathbf{A}\) when \(\theta \neq -1\). [3]
2. Find the rank of \(\mathbf{A}\) when \(\theta = -1\). [1]

Consider the system of equations \begin{align} x + 5y + z &= -1,
x - 2y - 2z &= 0,
2x + 3y + \theta z &= \theta. \end{align}

1. Solve the system of equations when \(\theta \neq -1\). [3]
2. Find the general solution when \(\theta = -1\). [3]
3. Show that if \(\theta = -1\) and \(\phi \neq -1\) then \(\mathbf{A}\mathbf{x} = \begin{pmatrix} -1 \\ 0 \\ \phi \end{pmatrix}\) has no solution. [2]

The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices \(\mathbf { M } _ { 1 }\) and \(\mathbf { M } _ { 2 }\), respectively, where $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & 2 \\ 1 & 4 & 7 & 8 \\ 1 & 7 & 11 & 13 \\ 1 & 2 & 5 & 5 \end{array} \right) , \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 2 & 0 & - 1 & - 1 \\ 5 & 1 & - 3 & - 3 \\ 3 & - 1 & - 1 & - 1 \\ 13 & - 1 & - 6 & - 6 \end{array} \right) .$$

1. Find a basis for \(R _ { 1 }\), the range space of \(\mathrm { T } _ { 1 }\).
2. Find a basis for \(K _ { 2 }\), the null space of \(\mathrm { T } _ { 2 }\), and hence show that \(K _ { 2 }\) is a subspace of \(R _ { 1 }\). The set of vectors which belong to \(R _ { 1 }\) but do not belong to \(K _ { 2 }\) is denoted by \(W\).
3. State whether \(W\) is a vector space, justifying your answer. The linear transformation \(\mathrm { T } _ { 3 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is the result of applying \(\mathrm { T } _ { 1 }\) and then \(\mathrm { T } _ { 2 }\), in that order.
4. Find the dimension of the null space of \(\mathrm { T } _ { 3 }\).

8

1. 1. Find the set of values of \(a\) for which the system of equations $$\begin{array} { r } x - 2 y - 2 z + 7 = 0

1

1. Find the set of values of \(k\) for which the system of equations $$\begin{aligned} x + 2 y + 3 z & = 1 \\ k x + 4 y + 6 z & = 0 \\ 7 x + 8 y + 9 z & = 3 \end{aligned}$$ has a unique solution.
2. Interpret the situation geometrically in the case where the system of equations does not have a unique solution.

14 Three planes have equations $$\begin{aligned} - x + a y & = 2 \\ 2 x + 3 y + z & = - 3 \\ x + b y + z & = c \end{aligned}$$ where \(a\), \(b\) and \(c\) are constants.

1. In the case where the planes do not intersect at a unique point,
   1. find \(b\) in terms of \(a\),
   2. find the value of \(c\) for which the planes form a sheaf.
2. In the case where \(b = a\) and \(c = 1\), find the coordinates of the point of intersection of the planes in terms of \(a\).

Show that the system of equations $$14x - 4y + 6z = 5,$$ $$x + y + kz = 3,$$ $$-21x + 6y - 9z = 14,$$ where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically. [4]

9 Three planes have equations \(k x + y - 2 z = 0\) \(2 x + 3 y - 6 z = - 5\) \(3 x - 2 y + 5 z = 1\) where \(k\) is a constant. Investigate the arrangement of the planes for each of the following cases. If in either case the planes meet at a unique point, find the coordinates of that point.

1. \(k = - 1\)
2. \(k = \frac { 2 } { 3 }\)

Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} where \(k\) is a real constant. The planes do not meet at a unique point.

1. Find the possible values of \(k\). [3 marks]
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]

1 Find the determinant of the matrix \(\left( \begin{array} { r r r } a & 4 & - 1 \\ 3 & a & 2 \\ a & 1 & 1 \end{array} \right)\).

3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } 2 & 1 & 3 \\ 1 & 2 & 1 \\ 1 & 1 & 3 \end{array} \right)\).

1. Find the value of the determinant of \(\mathbf { M }\).
2. State, giving a brief reason, whether \(\mathbf { M }\) is singular or non-singular.

5 A trading company deals in two goods. The formula used to estimate \(z\), the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is \(z = 0.9 x + \frac { 0.096 y } { x } - x ^ { 2 } y ^ { 2 }\),
where \(x\) and \(y\) are the masses, in thousands of tonnes, of the two goods. You are given that \(x > 0\) and \(y > 0\).

1. In the first week of trading, it was found that the values of \(x\) and \(y\) corresponded to the stationary value of \(z\). Determine the total cost to the company for this week.
2. For the second week, the company intends to make a small change in either \(x\) or \(y\) in order to reduce the total weekly cost. Determine whether the company should change \(x\) or \(y\). (You are not expected to say by how much the company should reduce its costs.)

2 Find the value of the constant \(k\) for which the system of equations $$\begin{aligned} 2 x - 3 y + 4 z & = 1 \\ 3 x - y & = 2 \\ x + 2 y + k z & = 1 \end{aligned}$$ does not have a unique solution. For this value of \(k\), solve the system of equations.

3

1. Find the inverse of the matrix $$\left( \begin{array} { r r r } 1 & 1 & a \\ 2 & - 1 & 2 \\ 3 & - 2 & 2 \end{array} \right)$$ where \(a \neq 4\).
   Show that when \(a = - 1\) the inverse is $$\frac { 1 } { 5 } \left( \begin{array} { r r r } 2 & 0 & 1 \\ 2 & 5 & - 4 \\ - 1 & 5 & - 3 \end{array} \right)$$
2. Solve, in terms of \(b\), the following system of equations. $$\begin{aligned} x + y - z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
3. Find the value of \(b\) for which the equations $$\begin{aligned} x + y + 4 z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$ have solutions. Give a geometrical interpretation of the solutions in this case. Section B (18 marks)

3

1. Find the value of \(a\) for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & a & 4 \\ 3 & - 2 & 2 \end{array} \right)$$ does not have an inverse.
   Assuming that \(a\) does not have this value, find the inverse of \(\mathbf { M }\) in terms of \(a\).
2. Hence solve the following system of equations. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
3. Find the value of \(b\) for which the following system of equations has a solution. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 6 y + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = b \end{aligned}$$ Find the general solution in this case and describe the solution geometrically.

1. The matrix \(\mathbf { M }\) is such that \(\mathbf { M } \left( \begin{array} { r r r } 1 & 0 & k \\ 2 & - 1 & 1 \end{array} \right) = \left( \begin{array} { l l l } 1 & - 2 & 0 \end{array} \right)\).
Find

• the matrix \(\mathbf { M }\),
• the value of the constant \(k\).
  [0pt]

2. Given that $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 2 \\ - 1 & 0 & 5 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } 2 & 4 \\ - k & 2 k \\ 3 & 0 \end{array} \right) , \text { where } k \text { is a constant }$$

1. find the matrix \(\mathbf { A B }\),
2. find the exact value of \(k\) for which \(\operatorname { det } ( \mathbf { A B } ) = 0\)

10 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } a & 8 & 10 \\ 2 & 1 & 2 \\ 4 & 3 & 6 \end{array} \right)\). The matrix \(\mathbf { B }\) is such that \(\mathbf { A B } = \left( \begin{array} { l l l } a & 6 & 1 \\ 1 & 1 & 0 \\ 1 & 3 & 0 \end{array} \right)\).

1. Show that \(\mathbf { A B }\) is non-singular.
2. Find \(( \mathbf { A B } ) ^ { - 1 }\).
3. Find \(\mathbf { B } ^ { - 1 }\).

You are given that the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{2a-a^2}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix}\), where \(a\) is a positive constant, represents the transformation \(R\) which is a reflection in 3-D.

1. State the plane of reflection of \(R\). [1]
2. Determine the value of \(a\). [3]
3. With reference to \(R\) explain why \(\mathbf{A}^2 = \mathbf{I}\), the \(3 \times 3\) identity matrix. [2]

3 You are given the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { 4 }\).
2. Describe the transformation that \(\mathbf { A }\) represents. The matrix \(\mathbf { B }\) represents a reflection in the plane \(x = 0\).
3. Write down the matrix \(\mathbf { B }\). The point \(P\) has coordinates (2, 3, 4). The point \(P ^ { \prime }\) is the image of \(P\) under the transformation represented by \(\mathbf { B }\).
4. Find the coordinates of \(P ^ { \prime }\).

6. $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & a \end{array} \right) \quad a \neq 1$$

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
   . The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { B }\). $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{array} \right)$$ The equation of \(l _ { 2 }\) is $$( \mathbf { r } - ( 12 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } ) ) \times ( - 6 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) = \mathbf { 0 }$$
2. Find a vector equation for the line \(l _ { 1 }\)

8 Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A } = \left( \begin{array} { r r r } 4 & - 1 & 1 \\ - 1 & 0 & - 3 \\ 1 & - 3 & 0 \end{array} \right)\). Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).

3. $$\mathbf { A } = \left( \begin{array} { l l l } 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{array} \right)$$

1. Find the eigenvalues of \(\mathbf { A }\).
2. Find a normalised eigenvector for each of the eigenvalues of \(\mathbf { A }\).
3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 3 & 3 & 0 \\ 0 & 2 & 2 \\ 1 & 3 & 4 \end{pmatrix}\).

1. Determine the characteristic equation of \(\mathbf{A}\). [3]
2. Hence verify that the eigenvalues of \(\mathbf{A}\) are 1, 2 and 6. [1]
3. For each eigenvalue of \(\mathbf{A}\) determine an associated eigenvector. [4]
4. Use the results of parts (b) and (c) to find \(\mathbf{A}^n\) as a single matrix, where \(n\) is a positive integer. [6]

6 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r r } a & 1 & 0 \\ 1 & 2 & 1 \\ - 1 & 3 & 4 \end{array} \right)\), where \(a \neq 1\). Find \(\mathbf { C } ^ { - 1 }\).

5 You are given that \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 4 \\ 3 & 2 & 5 \\ 4 & 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 1 & 0 & 2 \\ 14 & - 14 & 7 \\ - 5 & 7 & - 4 \end{array} \right)\).

1. Calculate AB.
2. Write down \(\mathbf { A } ^ { - 1 }\).

6. $$\mathbf { A } = \left( \begin{array} { r r r } x & 1 & 3 \\ 2 & 4 & x \\ - 4 & - 2 & - 1 \end{array} \right)$$

1. Show that \(\mathbf { A }\) is non-singular for all real values of \(x\).
2. Determine, in terms of \(x , \mathbf { A } ^ { - 1 }\)

5. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & k \\ - 1 & - 3 & 4 \\ 2 & 6 & - 8 \end{array} \right) \quad \text { where } k \text { is a constant }$$ Given that \(\mathbf { M }\) has a repeated eigenvalue, determine

1. the possible values of \(k\),
2. all corresponding eigenvalues of \(\mathbf { M }\) for each value of \(k\).

1. \(\mathbf { M } = \left( \begin{array} { r r r } 0 & 1 & 9 \\ 1 & 4 & k \\ 1 & 0 & - 3 \end{array} \right)\), where \(k\) is a constant.

Given that \(\left( \begin{array} { r } 7 \\ 19 \\ 1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),

1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { r } 7 \\ 19 \\ 1 \end{array} \right)\),
2. show that \(k = - 7\)
3. find the other two eigenvalues of the matrix \(\mathbf { M }\). The image of the vector \(\left( \begin{array} { c } p \\ q \\ r \end{array} \right)\) under the transformation represented by \(\mathbf { M }\) is \(\left( \begin{array} { r } - 6 \\ 21 \\ 5 \end{array} \right)\).
4. Find the values of the constants \(p , q\) and \(r\).

In this question you must show detailed reasoning. You are given that the matrix $\mathbf{M} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2}
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \end{pmatrix}$ represents a rotation in 3-D space.

1. Explain why it follows that \(\mathbf{M}\) has 1 as an eigenvalue. [2]
2. Find a vector equation for the axis of the rotation. [4]
3. Show that the characteristic equation of \(\mathbf{M}\) can be written as $$\lambda^3 - \lambda^2 + \lambda - 1 = 0.$$ [5]
4. Find the smallest positive integer \(n\) such that \(\mathbf{M}^n = \mathbf{I}\). [6]
5. Find the magnitude of the angle of the rotation which \(\mathbf{M}\) represents. Give your reasoning. [1]

5 Show that if \(a \neq 3\) then the system of equations $$\begin{aligned} & 2 x + 3 y + 4 z = - 5 \\ & 4 x + 5 y - z = 5 a + 15 \\ & 6 x + 8 y + a z = b - 2 a + 21 \end{aligned}$$ has a unique solution. Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.

3 The matrix \(\mathbf { A }\) is such that \(\operatorname { det } ( \mathbf { A } ) = 2\) Determine the value of \(\operatorname { det } \left( \mathbf { A } ^ { - 1 } \right)\) Circle your answer.
-2 \(- \frac { 1 } { 2 }\) \(\frac { 1 } { 2 }\) 2

1. Factorise \(\begin{vmatrix} 2u + h + x & x + h & x^2 + h^2 \\ 0 & a & -a^2 \\ a + b & b & b^2 \end{vmatrix}\) as fully as possible. [6 marks]
2. The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{bmatrix} 13 + x & x + 3 & x^2 + 9 \\ 0 & 5 & 25 \\ 8 & 3 & 9 \end{bmatrix}$$ Under the transformation represented by \(\mathbf{M}\), a solid of volume \(0.625 \text{m}^3\) becomes a solid of volume \(300 \text{m}^3\) Use your answer to part (a) to find the possible values of \(x\). [3 marks]

4 You are given that if \(\mathbf { M } = \left( \begin{array} { r r r } 4 & 0 & 1 \\ - 6 & 1 & 1 \\ 5 & 2 & 5 \end{array} \right)\) then \(\mathbf { M } ^ { - 1 } = \frac { 1 } { k } \left( \begin{array} { r r r } - 3 & - 2 & 1 \\ - 35 & - 15 & 10 \\ 17 & 8 & - 4 \end{array} \right)\).
Find the value of \(k\). Hence solve the following simultaneous equations. $$\begin{aligned} 4 x + z & = 9 \\ - 6 x + y + z & = 32 \\ 5 x + 2 y + 5 z & = 81 \end{aligned}$$

5 A matrix \(\mathbf { A }\) has eigenvalues \(- 1,1\) and 2 , with corresponding eigenvectors $$\left( \begin{array} { r } 0 \\ 1 \\ - 2 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) ,$$ respectively. Find \(\mathbf { A }\).

, \quad \mathbf { b } = \left( \begin{array} { l } 1
1
1 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 0
1
- 1 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 2
- 1
1 \end{array} \right)

The matrix A is \(\begin{pmatrix} -1 & 2 & 4 \\ 0 & -1 & -25 \\ -3 & 5 & -1 \end{pmatrix}\). Use the Cayley-Hamilton theorem to find A\(^{-1}\). [8]

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 0 & - 4 & - 3 \\ 0 & - 4 & 0 \end{array} \right)$$ Given that \(\mathbf { M }\) has exactly two distinct eigenvalues \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\) where \(\lambda _ { 1 } < \lambda _ { 2 }\)

1. determine a normalised eigenvector corresponding to the eigenvalue \(\lambda _ { 1 }\) The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 4 \\ - 1 \\ 0 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right)\), where \(\mu\) is a scalar parameter.
   The transformation \(T\) is represented by \(\mathbf { M }\).
   The line \(l _ { 1 }\) is transformed by \(T\) to the line \(l _ { 2 }\)
2. Determine a vector equation for \(l _ { 2 }\), giving your answer in the form \(\mathbf { r } \times \mathbf { b } = \mathbf { c }\) where \(\mathbf { b }\) and \(\mathbf { c }\) are constant vectors.

The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 4 & 8 & 0 \\ 0 & \lambda & -2 \\ 4 & 0 & \lambda \end{pmatrix}.$$

1. Show that there are two values of \(\lambda\) for which \(\mathbf{A}\) is singular. [4]
2. Given that \(\lambda = 3\),
   1. determine the adjugate matrix of \(\mathbf{A}\),
   2. determine the inverse matrix \(\mathbf{A}^{-1}\). [5]