4.03o Inverse 3x3 matrix

6 The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\). Find the corresponding eigenvalue. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).

4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 2 & 1 & 2 \\ 1 & - 1 & 1 \\ 2 & 2 & a \end{array} \right)\).

1. Show that \(\operatorname { det } \mathbf { A } = 6 - 3 a\).
2. State the value of \(a\) for which \(\mathbf { A }\) is singular.
3. Given that \(\mathbf { A }\) is non-singular find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).

8 Three transformations, \(T _ { A } , T _ { B }\) and \(T _ { C }\), are represented by the matrices \(A , B\) and \(\mathbf { C }\) respectively. You are given that \(\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right)\).

1. Find the matrix which represents the inverse transformation of \(T _ { A }\).
2. By considering matrix multiplication, determine whether \(T _ { A }\) followed by \(T _ { B }\) is the same transformation as \(T _ { B }\) followed by \(T _ { A }\). Transformations R and S are each defined as being the result of successive transformations, as specified in the table.

   Transformation	First transformation	followed by
   R	\(\mathrm { T } _ { \mathrm { A } }\) followed by \(\mathrm { T } _ { \mathrm { B } }\)	\(\mathrm { T } _ { \mathrm { C } }\)
   S	\(\mathrm { T } _ { \mathrm { A } }\)	\(\mathrm { T } _ { \mathrm { B } }\) followed by \(\mathrm { T } _ { \mathrm { C } }\)

3. Explain, using a property of matrix multiplication, why R and S are the same transformations. A quadrilateral, \(Q\), has vertices \(D , E , F\) and \(G\) in anticlockwise order from \(D\). Under transformation \(\mathrm { R } , Q ^ { \prime }\) s image, \(Q ^ { \prime }\), has vertices \(D ^ { \prime } , E ^ { \prime } , F ^ { \prime }\) and \(G ^ { \prime }\) (where \(D ^ { \prime }\) is the image of \(D\), etc). The area of \(Q\), in suitable units, is 5 . You are given that det \(\mathbf { C } = a ^ { 2 } + 1\) where \(a\) is a real constant.
4. 1. Determine the order of the vertices of \(Q ^ { \prime }\), starting anticlockwise from \(D ^ { \prime }\).
   2. Find, in terms of \(a\), the area of \(Q ^ { \prime }\).
   3. Explain whether the inverse transformation for R exists. Justify your answer.

3

1. You are given two matrices, A and B, where $$\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } - 1 & 2 \\ 2 & - 1 \end{array} \right)$$ Show that \(\mathbf { A B } = m \mathbf { I }\), where \(m\) is a constant to be determined.
2. You are given two matrices, \(\mathbf { C }\) and \(\mathbf { D }\), where $$\mathbf { C } = \left( \begin{array} { r r r } 2 & 1 & 5 \\ 1 & 1 & 3 \\ - 1 & 2 & 2 \end{array} \right) \text { and } \mathbf { D } = \left( \begin{array} { r r r } - 4 & 8 & - 2 \\ - 5 & 9 & - 1 \\ 3 & - 5 & 1 \end{array} \right)$$ Show that \(\mathbf { C } ^ { - 1 } = k \mathbf { D }\) where \(k\) is a constant to be determined.
3. The matrices \(\mathbf { E }\) and \(\mathbf { F }\) are given by \(\mathbf { E } = \left( \begin{array} { c c } k & k ^ { 2 } \\ 3 & 0 \end{array} \right)\) and \(\mathbf { F } = \binom { 2 } { k }\) where \(k\) is a constant. Determine any matrix \(\mathbf { F }\) for which \(\mathbf { E F } = \binom { - 2 k } { 6 }\).

7 You are given that \(a\) is a parameter which can take only real values.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c r } 2 & 4 & - 6 \\ - 3 & 10 - 4 a & 9 \\ 7 & 4 & 4 \end{array} \right)\).

1. Find an expression for the determinant of \(\mathbf { A }\) in terms of \(a\). You are given the following system of equations in \(x , y\) and \(z\). $$\begin{array} { r r } 2 x + & 4 y - 6 z = \\ - 3 x + & ( 10 - 4 a ) y + 9 z = \\ 7 x + & 4 y + 4 z = \\ 7 x + & 11 \end{array}$$ The system can be written in the form \(\mathbf { A } \left( \begin{array} { c } \mathrm { x } \\ \mathrm { y } \\ \mathrm { z } \end{array} \right) = \left( \begin{array} { r } 6 \\ - 9 \\ 11 \end{array} \right)\).
2. 1. In the case where \(\mathbf { A }\) is not singular, solve the given system of equations by using \(\mathbf { A } ^ { - 1 }\).
   2. In the case where \(\mathbf { A }\) is singular describe the configuration of the planes whose equations are the three equations of the system. The transformation represented by \(\mathbf { A }\) is denoted by T .
      A 3-D object of volume \(| 5 a - 20 |\) is transformed by T to a 3-D image.
3. 1. Determine the range of values of \(a\) for which the orientation of the image is the reverse of the orientation of the object.
   2. Determine the range of values of \(a\) for which the volume of the image is less than the volume of the object.

3 In this question you must show detailed reasoning. \(\mathbf { A }\) and \(\mathbf { B }\) are matrices such that \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 2 & 1 \\ - 1 & 1 \end{array} \right)\).

1. Find \(\mathbf { A B }\).
2. Given that \(\mathbf { A } = \left( \begin{array} { l l } \frac { 1 } { 3 } & 1 \\ 0 & 1 \end{array} \right)\), find \(\mathbf { B }\).

4

1. Find \(\mathbf { M } ^ { - 1 }\), where \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & 1 & 2 \\ - 2 & 1 & 2 \end{array} \right)\).
2. Hence find, in terms of the constant \(k\), the point of intersection of the planes $$\begin{aligned} x + 2 y + 3 z & = 19 \\ - x + y + 2 z & = 4 \\ - 2 x + y + 2 z & = k \end{aligned}$$
3. In this question you must show detailed reasoning. Find the acute angle between the planes \(x + 2 y + 3 z = 19\) and \(- x + y + 2 z = 4\).

8 A transformation T of the plane has matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } \cos \theta & 2 \cos \theta - \sin \theta \\ \sin \theta & 2 \sin \theta + \cos \theta \end{array} \right)\).

1. Show that T leaves areas unchanged for all values of \(\theta\).
2. Find the value of \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), for which the \(y\)-axis is an invariant line of T . The matrix \(\mathbf { N }\) is \(\left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).
3. 1. Find \(\mathbf { M N } ^ { - 1 }\).
   2. Hence describe fully a sequence of two transformations of the plane that is equivalent to T . \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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6 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 4 & - 9 \\ 1 & - 2 \end{array} \right)\).

1. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 + 3 n & - 9 n \\ n & 1 - 3 n \end{array} \right)\) for all positive integers \(n\).
2. A student thinks that this formula, when \(n = 0\) and \(n = - 1\), gives the identity matrix and the inverse matrix \(\mathbf { M } ^ { - 1 }\) respectively. Determine whether the student is correct.

2

1. The matrices \(\mathbf { M } = \left( \begin{array} { c c c } 0 & 1 & a \\ 1 & b & 0 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { c c } b & - 5 \\ - 1 & c \\ - 1 & 1 \end{array} \right)\) are such that \(\mathbf { M } \mathbf { N } = \mathbf { I }\).
   Find \(a , b\) and \(c\).
2. State with a reason whether or not \(\mathbf { N }\) is the inverse of \(\mathbf { M }\).

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 10 & 12 & - 8 \\ - 1 & 2 & 4 \\ 3 & 6 & 2 \end{array} \right)\).

1. In this question you must show detailed reasoning. Show that the characteristic equation of \(\mathbf { A }\) is \(- \lambda ^ { 3 } + 14 \lambda ^ { 2 } - 56 \lambda + 64 = 0\).
2. Use the Cayley-Hamilton theorem to determine \(\mathbf { A } ^ { - 1 }\). A matrix \(\mathbf { E }\) and a diagonal matrix \(\mathbf { D }\) are such that \(\mathbf { A } = \mathbf { E D E } ^ { - 1 }\). The elements in the diagonal of \(\mathbf { D }\) increase from top left to bottom right.
3. Determine the matrix \(\mathbf { D }\).

4 The set \(G\) is given by \(G = \{ \mathbf { M } : \mathbf { M }\) is a real \(2 \times 2\) matrix and det \(\mathbf { M } = 1 \}\).

1. Show that \(G\) forms a group under matrix multiplication, × . You may assume that matrix multiplication is associative.
2. The matrix \(\mathbf { A } _ { n }\) is defined by \(\mathbf { A } _ { n } = \left( \begin{array} { l l } 1 & 0 \\ n & 1 \end{array} \right)\) for any integer \(n\). The set \(S\) is defined by \(\mathrm { S } = \left\{ \mathrm { A } _ { \mathrm { n } } : \mathrm { n } \in \mathbb { Z } , \mathrm { n } \geqslant 0 \right\}\).
   1. Determine whether \(S\) is closed under × .
   2. Determine whether \(S\) is a subgroup of ( \(G , \times\) ).
3. 1. Find a subgroup of ( \(G , \times\) ) of order 2 .
   2. By considering the inverse of the non-identity element in any such subgroup, or otherwise, show that this is the only subgroup of ( \(G , \times\) ) of order 2. The set of all real \(2 \times 2\) matrices is denoted by \(H\).
4. With the help of an example, explain why ( \(H , \times\) ) is not a group.

5 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { l l } a & 0 \\ 2 & 3 \end{array} \right)\) where \(a\) is a constant and \(a \neq 3\).

1. Given that the acute angle between the directions of the eigenvectors of \(\mathbf { P }\) is \(\frac { 1 } { 4 } \pi\) radians, determine the possible values of \(a\).
2. You are given instead that \(\mathbf { P }\) satisfies the matrix equation \(\mathbf { I } = \mathbf { P } ^ { 2 } + r \mathbf { P }\) for some rational number \(r\).
   1. Use the Cayley-Hamilton theorem to determine the value of \(a\) and the corresponding value of \(r\).
   2. Hence show that \(\mathbf { P } ^ { 4 } = \mathbf { s } \mathbf { + t } \mathbf { t } \mathbf { P }\) where \(s\) and \(t\) are rational numbers to be determined. You should not calculate \(\mathbf { P } ^ { 4 }\).

4 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { r r r } 1 & 7 & 8 \\ - 6 & 12 & 12 \\ - 2 & 4 & 8 \end{array} \right)\).

1. Show that the characteristic equation of \(\mathbf { P }\) is \(- \lambda ^ { 3 } + 21 \lambda ^ { 2 } - 126 \lambda + 216 = 0\). You are given that the roots of this equation are 3,6 and 12 .
2. 1. Verify that \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)\) is an eigenvector of \(\mathbf { P }\), stating its associated eigenvalue.
   2. The vector \(\left( \begin{array} { l } x \\ y \\ z \end{array} \right)\) is an eigenvector of \(\mathbf { P }\) with eigenvalue 6. Given that \(z = 5\), find \(x\) and \(y\). You are given that \(\mathbf { P }\) can be expressed in the form \(\mathbf { E D E } ^ { - 1 }\), where \(\mathbf { E } = \left( \begin{array} { r r r } 3 & 2 & 1 \\ 1 & 2 & - 2 \\ 1 & 1 & 2 \end{array} \right)\) and \(\mathbf { D }\) is a diagonal matrix. The characteristic equation of \(\mathbf { E }\) is \(- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 15 \lambda + 9 = 0\).
3. 1. Use the Cayley-Hamilton theorem to express \(\mathbf { E } ^ { - 1 }\) in terms of positive powers of \(\mathbf { E }\).
   2. Hence find \(\mathbf { E } ^ { - 1 }\).
   3. By identifying the matrix \(\mathbf { D }\) and using \(\mathbf { P } = \mathbf { E D E } ^ { - 1 }\), determine \(\mathbf { P } ^ { 4 }\).

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

The matrix \(\mathbf { X }\) is such that \(\mathbf { A X } = \mathbf { B }\). Showing all your working, find the matrix \(\mathbf { X }\).

2. (a) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left( \begin{array} { c c } 3 & 4 \\ - 1 & - 2 \end{array} \right) , \quad \mathbf { B } = \binom { - 11 } { 7 }$$ Given that \(\mathbf { A X } = \mathbf { B }\), find the matrix \(\mathbf { X }\).
(b) (i) Find the \(2 \times 2\) matrix, \(\mathbf { T }\), which represents a reflection in the line \(y = - 2 x\).
(ii) The images of the points \(C ( 2,7 )\) and \(D ( 3,1 )\), under \(\mathbf { T }\), are \(E\) and \(F\) respectively. Find the coordinates of the midpoint of \(E F\).

2. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are such that \(\mathbf { A } = \left[ \begin{array} { c c } 2 & - 1 \\ 4 & - 7 \end{array} \right]\) and \(\mathbf { B } = \left[ \begin{array} { c c c } 2 & 0 & 9 \\ 4 & - 20 & 13 \end{array} \right]\).

1. Find the inverse of \(\mathbf { A }\).
2. Hence, find the matrix \(\mathbf { X }\), where \(\mathbf { A X } = \mathbf { B }\).

1. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & - 3 \\ 4 & - 2 & 1 \\ 3 & 5 & - 2 \end{array} \right)$$

1. Find \(\mathbf { M } ^ { - 1 }\) giving each element in exact form.
2. Solve the simultaneous equations $$\begin{array} { r } 2 x + y - 3 z = - 4 \\ 4 x - 2 y + z = 9 \\ 3 x + 5 y - 2 z = 5 \end{array}$$
3. Interpret the answer to part (b) geometrically.

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1. The population of chimpanzees in a particular country consists of juveniles and adults. Juvenile chimpanzees do not reproduce.

In a study, the numbers of juvenile and adult chimpanzees were estimated at the start of each year. A model for the population satisfies the matrix system $$\binom { J _ { n + 1 } } { A _ { n + 1 } } = \left( \begin{array} { c c } a & 0.15 \\ 0.08 & 0.82 \end{array} \right) \binom { J _ { n } } { A _ { n } } \quad n = 0,1,2 , \ldots$$ where \(a\) is a constant, and \(J _ { n }\) and \(A _ { n }\) are the respective numbers of juvenile and adult chimpanzees \(n\) years after the start of the study.

1. Interpret the meaning of the constant \(a\) in the context of the model. At the start of the study, the total number of chimpanzees in the country was estimated to be 64000 According to the model, after one year the number of juvenile chimpanzees is 15360 and the number of adult chimpanzees is 43008
2. 1. Find, in terms of \(a\) $$\left( \begin{array} { c c } a & 0.15 \\ 0.08 & 0.82 \end{array} \right) ^ { - 1 }$$
   2. Hence, or otherwise, find the value of \(a\).
   3. Calculate the change in the number of juvenile chimpanzees in the first year of the study, according to this model. Given that the number of juvenile chimpanzees is known to be in decline in the country,
3. comment on the short-term suitability of this model. A study of the population revealed that adult chimpanzees stop reproducing at the age of 40 years.
4. Refine the matrix system for the model to reflect this information, giving a reason for your answer.
   (There is no need to estimate any unknown values for the refined model, but any known values should be made clear.)

$$\mathbf { A } = \left( \begin{array} { c c } 2 & a \\ a - 4 & b \end{array} \right)$$ where \(a\) and \(b\) are non-zero constants.
Given that the matrix \(\mathbf { A }\) is self-inverse,

1. determine the value of \(b\) and the possible values for \(a\). The matrix \(\mathbf { A }\) represents a linear transformation \(M\).
   Using the smaller value of \(a\) from part (a),
2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line.
   (ii) $$\mathbf { P } = \left( \begin{array} { c c } p & 2 p \\ - 1 & 3 p \end{array} \right)$$ where \(p\) is a positive constant.
   The matrix \(\mathbf { P }\) represents a linear transformation \(U\).
   The triangle \(T\) has vertices at the points with coordinates ( 1,2 ), ( 3,2 ) and ( 2,5 ). The area of the image of \(T\) under the linear transformation \(U\) is 15
3. Determine the value of \(p\). The transformation \(V\) consists of a stretch scale factor 3 parallel to the \(x\)-axis with the \(y\)-axis invariant followed by a stretch scale factor - 2 parallel to the \(y\)-axis with the \(x\)-axis invariant. The transformation \(V\) is represented by the matrix \(\mathbf { Q }\).
4. Write down the matrix \(\mathbf { Q }\). Given that \(U\) followed by \(V\) is the transformation \(W\), which is represented by the matrix \(\mathbf { R }\), (c) find the matrix \(\mathbf { R }\).

4. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 4 \\ k & 2 & - 2 \\ 4 & 1 & - 2 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r r } k - 7 & 6 & - 10 \\ 2 & - 20 & 24 \\ - 3 & 2 & - 1 \end{array} \right)$$ where \(k\) is a constant.

1. Determine, in simplest form in terms of \(k\), the matrix \(\mathbf { M N }\).
2. Given that \(k = 5\)
   1. write down \(\mathbf { M N }\)
   2. hence write down \(\mathbf { M } ^ { - 1 }\)
3. Solve the simultaneous equations $$\begin{aligned} & 2 x + y + 4 z = 2 \\ & 5 x + 2 y - 2 z = 3 \\ & 4 x + y - 2 z = - 1 \end{aligned}$$
4. Interpret the answer to part (c) geometrically.

1. $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 1 \\ 7 & 2 \\ - 5 & 8 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 3 & 2 \\ - 1 & 6 & 5 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { r r r } - 5 & 2 & 1 \\ 4 & 3 & 8 \\ - 6 & 11 & 2 \end{array} \right)$$ Given that \(\mathbf { I }\) is the \(3 \times 3\) identity matrix,

1. 1. show that there is an integer \(k\) for which $$\mathbf { A B } - 3 \mathbf { C } + k \mathbf { I } = \mathbf { 0 }$$ stating the value of \(k\)
   2. explain why there can be no constant \(m\) such that $$\mathbf { B A } - 3 \mathbf { C } + m \mathbf { I } = \mathbf { 0 }$$
2. 1. Show how the matrix \(\mathbf { C }\) can be used to solve the simultaneous equations $$\begin{aligned} - 5 x + 2 y + z & = - 14 \\ 4 x + 3 y + 8 z & = 3 \\ - 6 x + 11 y + 2 z & = 7 \end{aligned}$$
   2. Hence use your calculator to solve these equations.

4. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & - 2 & - 7 \\ 3 & k & 2 \\ 1 & 1 & 4 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { c c c } 4 k - 2 & 1 & 7 k - 4 \\ - 10 & 3 & - 19 \\ 3 - k & - 1 & 6 - k \end{array} \right)$$ where \(k\) is a constant.

1. Determine the value of the constant \(c\) for which $$\mathbf { A B } = ( 3 k + c ) \mathbf { I }$$
2. Hence determine the value of \(k\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Given that \(\mathbf { A } ^ { - 1 }\) does exist,
3. write down \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
4. Use the answer to part (c) to solve the simultaneous equations $$\begin{aligned} - x - 2 y - 7 z & = 10 \\ 3 x + k y + 2 z & = 3 \\ x + y + 4 z & = 1 \end{aligned}$$ giving the values of \(x , y\) and \(z\) in simplest form in terms of \(k\).

4. $$\mathbf { A } = \left( \begin{array} { r r } 1 & 1 \\ - 2 & 4 \end{array} \right)$$ Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { D } = \mathbf { P } ^ { - 1 } \mathbf { A P }\)

$$A = \left( \begin{array} { r r } 1 & - 2 \\ 1 & 4 \end{array} \right)$$

1. Show that the characteristic equation for \(\mathbf { A }\) is \(\lambda ^ { 2 } - 5 \lambda + 6 = 0\)
2. Use the Cayley-Hamilton theorem to find integers \(p\) and \(q\) such that $$\mathbf { A } ^ { 3 } = p \mathbf { A } + q \mathbf { I }$$ (ii) Given that the \(2 \times 2\) matrix \(\mathbf { M }\) has eigenvalues \(- 1 + \mathrm { i }\) and \(- 1 - \mathrm { i }\), with eigenvectors \(\binom { 1 } { 2 - \mathrm { i } }\) and \(\binom { 1 } { 2 + \mathrm { i } }\) respectively, find the matrix \(\mathbf { M }\).