4.03j Determinant 3x3: calculation

3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 3 & 0 \\ 3 & - 2 & - 1 \\ 0 & - 1 & 1 \end{array} \right)\).

1. Show that the characteristic equation of \(\mathbf { M }\) is $$\lambda ^ { 3 } - 13 \lambda + 12 = 0 .$$
2. Find the eigenvalues and corresponding eigenvectors of \(\mathbf { M }\).
3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { M } ^ { n } = \mathbf { P D P } ^ { - 1 } .$$ (You are not required to calculate \(\mathbf { P } ^ { - 1 }\).)

3

1. 1. Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l } 6 & - 3 \\ 4 & - 1 \end{array} \right)$$
   2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
2. 1. The \(3 \times 3\) matrix \(\mathbf { B }\) has characteristic equation $$\lambda ^ { 3 } - 4 \lambda ^ { 2 } - 3 \lambda - 10 = 0$$ Show that 5 is an eigenvalue of \(\mathbf { B }\). Show that \(\mathbf { B }\) has no other real eigenvalues.
   2. An eigenvector corresponding to the eigenvalue 5 is \(\left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right)\). Evaluate \(\mathbf { B } \left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right)\) and \(\mathbf { B } ^ { 2 } \left( \begin{array} { r } 4 \\ - 2 \\ - 8 \end{array} \right)\).
      Solve the equation \(\mathbf { B } \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } - 20 \\ 10 \\ 40 \end{array} \right)\) for \(x , y , z\).
   3. Show that \(\mathbf { B } ^ { 4 } = 19 \mathbf { B } ^ { 2 } + 22 \mathbf { B } + 40 \mathbf { I }\).

3 This question concerns the matrix \(\mathbf { M }\) where \(\mathbf { M } = \left( \begin{array} { r r r } 5 & - 1 & 3 \\ 4 & - 3 & - 2 \\ 2 & 1 & 4 \end{array} \right)\).

1. Obtain the characteristic equation of \(\mathbf { M }\). Find the eigenvalues of \(\mathbf { M }\). These eigenvalues are denoted by \(\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }\), where \(\lambda _ { 1 } < \lambda _ { 2 } < \lambda _ { 3 }\).
2. Verify that an eigenvector corresponding to \(\lambda _ { 1 }\) is \(\left( \begin{array} { r } 1 \\ 3 \\ - 1 \end{array} \right)\) and that an eigenvector corresponding to \(\lambda _ { 2 }\) is \(\left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right)\). Find an eigenvector of the form \(\left( \begin{array} { l } a \\ 1 \\ c \end{array} \right)\) corresponding to \(\lambda _ { 3 }\).
3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { P D P } ^ { - 1 }\). (You are not required to calculate \(\mathbf { P } ^ { - 1 }\).) Hence write down an expression for \(\mathbf { M } ^ { 4 }\) in terms of \(\mathbf { P }\) and a diagonal matrix. You should give the elements of the diagonal matrix explicitly.
4. Use the Cayley-Hamilton theorem to obtain an expression for \(\mathbf { M } ^ { 4 }\) as a linear combination of \(\mathbf { M }\) and \(\mathbf { M } ^ { 2 }\).

3

1. Find the value of \(k\) for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & - 1 & k \\ 5 & 4 & 6 \\ 3 & 2 & 4 \end{array} \right)$$ does not have an inverse.
   Assuming that \(k\) does not take this value, find the inverse of \(\mathbf { M }\) in terms of \(k\).
2. In the case \(k = 3\), evaluate $$\mathbf { M } \left( \begin{array} { r } - 3 \\ 3 \\ 1 \end{array} \right)$$
3. State the significance of what you have found in part (ii).
4. Find the value of \(t\) for which the system of equations $$\begin{array} { r } x - y + 3 z = t \\ 5 x + 4 y + 6 z = 1 \\ 3 x + 2 y + 4 z = 0 \end{array}$$ has solutions. Find the general solution in this case and describe the solution geometrically.

5 By using the determinant of an appropriate matrix, or otherwise, find the value of \(k\) for which the simultaneous equations $$\begin{aligned} 2 x - y + z & = 7 \\ 3 y + z & = 4 \\ x + k y + k z & = 5 \end{aligned}$$ do not have a unique solution for \(x , y\) and \(z\).

9 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 \\ 3 & a & 1 \\ 4 & 2 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
2. Hence find the values of \(a\) for which \(\mathbf { M } ^ { - 1 }\) does not exist.
3. Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k \\ & 3 x + 6 y + z = 0 \\ & 4 x + 2 y + z = k \end{aligned}$$ where \(k\) is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
4. Show that \(\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }\).
5. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
6. Show that \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).

\(\mathbf { 9 }\) The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 2 & 9 \\ 2 & a & 3 \\ 1 & 0 & - 1 \end{array} \right)\).

1. Find the determinant of \(\mathbf { X }\) in terms of \(a\).
2. Hence find the values of \(a\) for which \(\mathbf { X }\) is singular.
3. Given that \(\mathbf { X }\) is non-singular, find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\).

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

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\).
2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular.
3. State, giving a brief reason in each case, whether the simultaneous equations $$\begin{aligned} a x + y + z & = 2 a \\ x + a y + z & = - 1 \\ x + y + 2 z & = - 1 \end{aligned}$$ have any solutions when
   1. \(a = 0\),
   2. \(a = 1\).

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

1. Find the determinant of \(\mathbf { D }\) in terms of \(a\).
2. Three simultaneous equations are shown below. $$\begin{array} { r } a x + 2 y - z = 0 \\ 2 x + a y + z = a \\ x + y + a z = a \end{array}$$ For each of the following values of \(a\), determine whether or not there is a unique solution. If the solution is not unique, determine whether the equations are consistent or inconsistent.
   1. \(\quad a = 3\)
   2. \(a = 2\)
   3. \(\quad a = 0\) \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

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

9 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l l } 1 & 3 & 4 \\ 2 & a & 3 \\ 0 & 1 & a \end{array} \right)\).

1. Find the values of \(a\) for which \(\mathbf { D }\) is singular.
2. Three simultaneous equations are shown below. $$\begin{array} { r } x + 3 y + 4 z = 3 \\ 2 x + a y + 3 z = 2 \\ y + a z = 0 \end{array}$$ For each of the following values of \(a\), determine whether or not there is a unique solution. If a unique solution does not exist, determine whether the equations are consistent or inconsistent.
   1. \(a = 3\)
   2. \(a = 1\)

9

1. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 3 & - 2 \\ 0 & a & 5 \\ 1 & 2 & 1 \end{array} \right)\). Show that the determinant of \(\mathbf { X }\) is \(a ^ { 2 } - 8 a + 15\).
2. Explain briefly why the equations $$\begin{array} { r } 3 x + 3 y - 2 z = 1 \\ 3 y + 5 z = 5 \\ x + 2 y + z = 2 \end{array}$$ do not have a unique solution and determine whether these equations are consistent or inconsistent.
3. Use an algebraic method to find the square roots of the complex number \(9 + 40 \mathrm { i }\).
4. Show that \(9 + 40 \mathrm { i }\) is a root of the quadratic equation \(z ^ { 2 } - 18 z + 1681 = 0\).
5. By using the substitution \(z = \frac { 1 } { u ^ { 2 } }\), find the roots of the equation \(1681 u ^ { 4 } - 18 u ^ { 2 } + 1 = 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

9 The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 4 \\ 1 & 5 & - 1 \\ 2 & 1 & 5 \end{array} \right)$$ has eigenvalues \(1,5,7\). Find a set of corresponding eigenvectors. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\).
[0pt] [The evaluation of \(\mathbf { P } ^ { - 1 }\) is not required.]
Determine the set of values of the real constant \(k\) such that \(k ^ { n } \mathbf { A } ^ { n }\) tends to the zero matrix as \(n \rightarrow \infty\).

One of the eigenvalues of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } 3 & - 4 & 2 \\ - 4 & \alpha & 6 \\ 2 & 6 & - 2 \end{array} \right)$$ is - 9 . Find the value of \(\alpha\). Find

1. the other two eigenvalues, \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\), of \(\mathbf { M }\), where \(\lambda _ { 1 } > \lambda _ { 2 }\),
2. corresponding eigenvectors for all three eigenvalues of \(\mathbf { M }\). It is given that \(\mathbf { x } = a \mathbf { e } _ { 1 } + b \mathbf { e } _ { 2 }\), where \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) are eigenvectors of \(\mathbf { M }\) corresponding to the eigenvalues \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\) respectively, and \(a\) and \(b\) are scalar constants. Show that \(\mathbf { M x } = p \mathbf { e } _ { 1 } + q \mathbf { e } _ { 2 }\), expressing \(p\) and \(q\) in terms of \(a\) and \(b\). {www.cie.org.uk} after the live examination series. }

10 The vectors \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 } , \mathbf { b } _ { 4 }\) are defined as follows: $$\mathbf { b } _ { 1 } = \left( \begin{array} { c } 1 \\ 0 \\ 0 \\ 0 \end{array} \right) , \quad \mathbf { b } _ { 2 } = \left( \begin{array} { c } 1 \\ 1 \\ 0 \\ 0 \end{array} \right) , \quad \mathbf { b } _ { 3 } = \left( \begin{array} { c } 1 \\ 1 \\ 1 \\ 0 \end{array} \right) , \quad \mathbf { b } _ { 4 } = \left( \begin{array} { c } 1 \\ 1 \\ 1 \\ 1 \end{array} \right) .$$ The linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 }\) is denoted by \(V _ { 1 }\) and the linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 4 }\) is denoted by \(V _ { 2 }\).

1. Give a reason why \(V _ { 1 } \cup V _ { 2 }\) is not a linear space.
2. State the dimension of the linear space \(V _ { 1 } \cap V _ { 2 }\) and write down a basis. Consider now the set \(V _ { 3 }\) of all vectors of the form \(q \mathbf { b } _ { 2 } + r \mathbf { b } _ { 3 } + s \mathbf { b } _ { 4 }\), where \(q , r , s\) are real numbers. Show that \(V _ { 3 }\) is a linear space, and show also that it has dimension 3 . Determine whether each of the vectors $$\left( \begin{array} { l } 4 \\ 4 \\ 2 \\ 5 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { l } 5 \\ 4 \\ 2 \\ 5 \end{array} \right)$$ belongs to \(V _ { 3 }\) and justify your conclusions.

11 Find the eigenvalues of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 1 & 4 \\ 1 & 1 & - 1 \\ 2 & 1 & 1 \end{array} \right)$$ and corresponding eigenvectors. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \mathbf { A } - k \mathbf { I } ,$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix and \(k\) is a real number. Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { B } ^ { 3 } = \mathbf { P D } \mathbf { P } ^ { - 1 } .$$

8 The vector \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf { B }\), with corresponding eigenvalue \(\mu\). Show that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with corresponding eigenvalue \(\lambda \mu\). State the eigenvalues of the matrix \(\mathbf { C }\), where $$\mathbf { C } = \left( \begin{array} { r r r } - 1 & - 1 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 2 \end{array} \right) ,$$ and find corresponding eigenvectors. Show that \(\left( \begin{array} { l } 1 \\ 6 \\ 3 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { D }\), where $$\mathbf { D } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ - 6 & - 3 & 4 \\ - 9 & - 3 & 7 \end{array} \right) ,$$ and state the corresponding eigenvalue. Hence state an eigenvector of the matrix CD and give the corresponding eigenvalue.

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 \(a\).

3 A transformation T is represented by the matrix \(\mathbf { N } = \left( \begin{array} { l l l } a & 4 & 2 \\ 5 & 1 & 0 \\ 3 & 6 & 3 \end{array} \right)\), where \(a\) is a constant.

1. Find \(\mathbf { N } ^ { 2 }\) in terms of \(a\).
2. Find det \(\mathbf { N }\) in terms of \(a\). The value of \(a\) is 13 to the nearest integer.
   A shape \(S _ { 1 }\) has volume 11.6 to 1 decimal place. Shape \(S _ { 1 }\) is mapped to shape \(S _ { 2 }\) by the transformation T . A student claims that the volume of \(S _ { 2 }\) is less than 400 .
3. Comment on the student's claim.

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.

8 The equations of three planes are $$\begin{array} { r } 2 x + y + 3 z = 3 \\ 3 x - y - 2 z = 2 \\ - 4 x + 3 y + 7 z = k \end{array}$$ where \(k\) is a constant.

1. By considering a suitable determinant, show that the planes do not meet at a single point.
2. Given that the planes form a sheaf, determine the value of \(k\).

5

1. Find the volume scale factor of the transformation with associated matrix \(\left( \begin{array} { r r r } 1 & 2 & 0 \\ 0 & 3 & - 1 \\ - 1 & 0 & 2 \end{array} \right)\).
2. The transformations S and T of the plane have associated \(2 \times 2\) matrices \(\mathbf { P }\) and \(\mathbf { Q }\) respectively.
   1. Write down an expression for the associated matrix of the combined transformation S followed by T. The determinant of \(\mathbf { P }\) is 3 and \(\mathbf { Q } = \left( \begin{array} { r r } k & 3 \\ - 1 & 2 \end{array} \right)\), where \(k\) is a constant.
   2. Given that this combined transformation preserves both orientation and area, determine the value of \(k\).

7 Three planes have equations $$\begin{array} { r } x + 2 y - 3 z = 0 \\ - x + 3 y - 2 z = 0 \\ x - 2 y + k z = k \end{array}$$ where \(k\) is a constant.

1. For the case \(k = 0\), the origin lies on all three planes. Use a determinant to explain whether there are any other points that lie on all three planes in this case.
2. You are now given that \(k = 1\).
   1. Show that there are no points that lie on all three planes.
   2. Describe the geometrical arrangement of the three planes.

4 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r r } 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)\).

1. 1. Calculate \(\operatorname { det } \mathbf { M }\).
   2. State two geometrical consequences of this value for the transformation associated with \(\mathbf { M }\).
2. Describe fully the transformation associated with \(\mathbf { M }\).