SL Paper 1
Consider the simultaneous linear equations
\(x + z = - 1\)
\(3x + y + 2z = 1\)
\(2x + ay - z = b\)
where \(a\) and \(b\) are constants.
Using row reduction, find the solutions in terms of \(a\) and \(b\) when \(a\) ≠ 3 .
Explain why the equations have no unique solution when \(a\) = 3.
Find all the solutions to the equations when \(a\) = 3, \(b\) = 10 in the form r = s + \(\lambda \)t.
Consider the matrix M = \(\left[ {\begin{array}{*{20}{c}}
2 \\
{ - 1}
\end{array}\,\,\,\begin{array}{*{20}{c}}
{ - 4} \\
{ - 1}
\end{array}} \right]\).
Show that the linear transformation represented by M transforms any point on the line \(y = x\) to a point on the same line.
Explain what happens to points on the line \(4y + x = 0\) when they are transformed by M.
State the two eigenvalues of M.
State two eigenvectors of M which correspond to the two eigenvalues.
A matrix M is called idempotent if M\(^2 = \) M.
The idempotent matrix N has the form
N \( = \left( {\begin{array}{*{20}{c}} a&{ - 2a} \\ a&{ - 2a} \end{array}} \right)\)
where \(a \ne 0\).
(i) Explain why M is a square matrix.
(ii) Find the set of possible values of det(M).
(i) Find the value of \(a\).
(ii) Find the eigenvalues of N.
(iii) Find corresponding eigenvectors.
Let A2 = 2A + I where A is a 2 × 2 matrix.
Show that A4 = 12A + 5I.
Let B = \(\left[ {\begin{array}{*{20}{c}}
4&2 \\
1&{ - 3}
\end{array}} \right]\).
Given that B2 – B – 4I = \(\left[ {\begin{array}{*{20}{c}}
k&0 \\
0&k
\end{array}} \right]\), find the value of \(k\).
Consider the system of equations
\[\left[ {\begin{array}{*{20}{l}} 1&2&1&3 \\ 2&1&3&1 \\ 5&1&8&0 \\ 3&3&4&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {{x_3}} \\ {{x_4}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2 \\ 3 \\ \lambda \\ \mu \end{array}} \right]\]
Determine the value of \(\lambda \) and the value of \(\mu \) for which the equations are consistent.
For these values of \(\lambda \) and \(\mu \), solve the equations.
State the rank of the matrix of coefficients, justifying your answer.
The non-zero vectors v1, v2, v3 form an orthogonal set of vectors in \({\mathbb{R}^3}\).
By considering \({\alpha _1}\)v\(_1 + {\alpha _2}\)v\(_2 + {\alpha _3}\)v\(_3 = 0\), show that v\(_1\), v\(_2\), v\(_3\) are linearly independent.
Explain briefly why v\(_1\), v\(_2\), v\(_3\) form a basis for vectors in \({\mathbb{R}^3}\).
Show that the vectors
\[\left[ {\begin{array}{*{20}{c}} 1 \\ 0 \\ 1 \end{array}} \right];{\text{ }}\left[ {\begin{array}{*{20}{c}} { - 1} \\ 1 \\ 1 \end{array}} \right];{\text{ }}\left[ {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right]\]
form an orthogonal basis.
Express the vector
\[\left[ {\begin{array}{*{20}{c}} 2 \\ 8 \\ 0 \end{array}} \right]\]
as a linear combination of these vectors.
In this question, \(x\), \(y\) and \(z\) denote the coordinates of a point in three-dimensional Euclidean space with respect to fixed rectangular axes with origin O. The vector space of position vectors relative to O is denoted by \({\mathbb{R}^3}\).
Explain why the set of position vectors of points whose coordinates satisfy \(x - y - z = 1\) does not form a vector subspace of \({\mathbb{R}^3}\).
(i) Show that the set of position vectors of points whose coordinates satisfy \(x - y - z = 0\) forms a vector subspace, \(V\), of \({\mathbb{R}^3}\).
(ii) Determine an orthogonal basis for \(V\) of which one member is \(\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right)\).
(iii) Augment this basis with an orthogonal vector to form a basis for \({\mathbb{R}^3}\).
(iv) Express the position vector of the point with coordinates \((4,{\text{ }}0,{\text{ }} - 2)\) as a linear combination of these basis vectors.
The matrix A is given by A = \(\left( {\begin{array}{*{20}{c}}1&2&1\\1&1&2\\2&3&1\end{array}} \right)\).
(a) Given that A\(^3\) can be expressed in the form A\(^3 = a\)A\(^2 = b\)A \( + c\)I, determine the values of the constants \(a\), \(b\), \(c\).
(b) (i) Hence express A\(^{ - 1}\) in the form A\(^{ - 1} = d\)A\(^2 = e\)A \( + f\)I where \(d,{\text{ }}e,{\text{ }}f \in \mathbb{Q}\).
(ii) Use this result to determine A\(^{ - 1}\).
A transformation \(T\) is a linear mapping from \({\mathbb{R}^3}\) to \({\mathbb{R}^4}\), represented by the matrix
\[M = \left( {\begin{array}{*{20}{c}} 1&2&1 \\ 2&7&5 \\ { - 3}&1&4 \\ 1&5&4 \end{array}} \right)\]
(i) Find the row rank of \(M\).
(ii) Hence or otherwise find the kernel of \(T\).
(i) State the column rank of \(M\).
(ii) Find the basis for the range of this transformation.
Let S be the set of matrices given by
\(\left[ \begin{array}{l}
a\\
c
\end{array} \right.\left. \begin{array}{l}
b\\
d
\end{array} \right]\) ; \(a,b,c,d \in \mathbb{R}\), \(ad - bc = 1\)
The relation \(R\) is defined on \(S\) as follows. Given \(\boldsymbol{A}\) , \(\boldsymbol{B} \in S\) , \(\boldsymbol{ARB}\) if and only if there exists \(\boldsymbol{X} \in S\) such that \(\boldsymbol{A} = \boldsymbol{BX}\) .
Show that \(R\) is an equivalence relation.
The relationship between \(a\) , \(b\) , \(c\) and \(d\) is changed to \(ad - bc = n\) . State, with a reason, whether or not there are any non-zero values of \(n\) , other than \(1\), for which \(R\) is an equivalence relation.
The matrix M is defined by M = \(\left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\).
The eigenvalues of M are denoted by \({\lambda _1},{\text{ }}{\lambda _2}\).
(a) Show that \({\lambda _1} + {\lambda _2} = a + d\) and \({\lambda _1}{\lambda _2} = \det \)(M).
(b) Given that \(a + b = c + d = 1\), show that 1 is an eigenvalue of M.
(c) Find eigenvectors for the matrix \(\left( {\begin{array}{*{20}{c}}2&{ - 1}\\3&{ - 2}\end{array}} \right)\).
The matrix \(\boldsymbol{A}\) is given by \[\boldsymbol{A} = \left( {\begin{array}{*{20}{c}}
0&1&0\\
2&4&1\\
4&{ - 11}&{ - 2}
\end{array}} \right) .\]
(i) Find the matrices \({\boldsymbol{A}^2}\) and \({\boldsymbol{A}^3}\) , and verify that \({{\boldsymbol{A}}^3} = 2{{\boldsymbol{A}}^2} - {\boldsymbol{A}}\) .
(ii) Deduce that \({{\boldsymbol{A}}^4} = 3{{\boldsymbol{A}}^2} - 2{\boldsymbol{A}}\) .
(i) Suggest a similar expression for \({\boldsymbol{A}^n}\) in terms of \(\boldsymbol{A}\) and \({\boldsymbol{A}^2}\) , valid for \(n \ge 3\) .
(ii) Use mathematical induction to prove the validity of your suggestion.
Consider the system of equations \[\left( {\begin{array}{*{20}{c}}
1&{ - 1}&2\\
2&2&{ - 1}\\
3&5&{ - 4}\\
3&1&1
\end{array}} \right)\left( \begin{array}{l}
x\\
y\\
z
\end{array} \right) = \left( \begin{array}{l}
5\\
3\\
1\\
k
\end{array} \right) .\]
By reducing the augmented matrix to row echelon form,
(i) find the rank of the coefficient matrix;
(ii) find the value of \(k\) for which the system has a solution.
For this value of \(k\) , determine the solution.
Show that the following vectors form a basis for the vector space \({\mathbb{R}^3}\) .\[\left( \begin{array}{l}
1\\
2\\
3
\end{array} \right);\left( \begin{array}{l}
2\\
3\\
1
\end{array} \right);\left( \begin{array}{l}
5\\
2\\
5
\end{array} \right)\]
Express the following vector as a linear combination of the above vectors.\[\left( \begin{array}{l}
12\\
14\\
16
\end{array} \right)\]
The set \(S\) contains the eight matrices of the form\[\left( {\begin{array}{*{20}{c}}
a&0&0\\
0&b&0\\
0&0&c
\end{array}} \right)\]where \(a\), \(b\), \(c\) can each take one of the values \( + 1\) or \( - 1\) .
Show that any matrix of this form is its own inverse.
Show that \(S\) forms an Abelian group under matrix multiplication.
Giving a reason, state whether or not this group is cyclic.
By considering the images of the points (1, 0) and (0, 1),
determine the 2 × 2 matrix P which represents a reflection in the line \(y = \left( {{\text{tan}}\,\theta } \right)x\).
determine the 2 × 2 matrix Q which represents an anticlockwise rotation of θ about the origin.
Describe the transformation represented by the matrix PQ.
A matrix M is said to be orthogonal if M TM = I where I is the identity. Show that Q is orthogonal.
The transformations T1, T2, T3, T4, in the plane are defined as follows:
T1 : A rotation of 360° about the origin
T2 : An anticlockwise rotation of 270° about the origin
T3 : A rotation of 180° about the origin
T4 : An anticlockwise rotation of 90° about the origin.
The transformation T5 is defined as a reflection in the \(x\)-axis.
The transformation T is defined as the composition of T3 followed by T5 followed by T4.
Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the operation of composition of transformations.
Show that T1, T2, T3, T4 under the operation of composition of transformations form a group. Associativity may be assumed.
Show that this group is cyclic.
Write down the 2 × 2 matrices representing T3, T4 and T5.
Find the 2 × 2 matrix representing T.
Give a geometric description of the transformation T.