4.03o Inverse 3x3 matrix

1. 1. The matrix \(\mathbf { A }\) is defined by

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

1. Determine the value of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
2. determine \(\mathbf { A } ^ { - 1 }\) in terms of \(k\), giving your answer in simplest form.
   (ii) The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left( \begin{array} { l l } p & 0 \\ 0 & q \end{array} \right)$$ where \(p\) and \(q\) are integers.
   State the value of \(p\) and the value of \(q\) when \(\mathbf { B }\) represents
3. an enlargement about the origin with scale factor - 2
4. a reflection in the \(y\)-axis.

1. $$\mathbf { A } = \left( \begin{array} { r r } 3 & a \\ - 2 & - 2 \end{array} \right)$$ where \(a\) is a non-zero constant and \(a \neq 3\)

1. Determine \(\mathbf { A } ^ { - 1 }\) giving your answer in terms of \(a\). Given that \(\mathbf { A } + \mathbf { A } ^ { - 1 } = \mathbf { I }\) where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. determine the value of \(a\).

1. Given that

$$\mathbf { A } = \left( \begin{array} { c c } k & 3 \\ - 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$

1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
   

7. Given that \(\mathbf { X } = \left( \begin{array} { c c } 2 & a \\ - 1 & - 1 \end{array} \right)\), where \(a\) is a constant, and \(a \neq 2\),

1. find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { X } + \mathbf { X } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(a\).

6. \(\mathbf { X } = \left( \begin{array} { l l } 1 & a \\ 3 & 2 \end{array} \right)\), where \(a\) is a constant.

1. Find the value of \(a\) for which the matrix \(\mathbf { X }\) is singular. $$\mathbf { Y } = \left( \begin{array} { r r } 1 & - 1 \\ 3 & 2 \end{array} \right)$$
2. Find \(\mathbf { Y } ^ { - 1 }\). The transformation represented by \(\mathbf { Y }\) maps the point \(A\) onto the point \(B\).
   Given that \(B\) has coordinates ( \(1 - \lambda , 7 \lambda - 2\) ), where \(\lambda\) is a constant,
3. find, in terms of \(\lambda\), the coordinates of point \(A\).

7. $$\mathbf { P } = \left( \begin{array} { c c } 3 a & - 2 a \\ - b & 2 b \end{array} \right) , \quad \mathbf { M } = \left( \begin{array} { c c } - 6 a & 7 a \\ 2 b & - b \end{array} \right)$$ where \(a\) and \(b\) are non-zero constants.

1. Find \(\mathbf { P } ^ { - 1 }\), leaving your answer in terms of \(a\) and \(b\). Given that $$\mathbf { M } = \mathbf { P Q }$$
2. find the matrix \(\mathbf { Q }\), giving your answer in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-19_95_77_2617_1804}

7. \(\mathbf { A } = \left( \begin{array} { r r } a & - 2 \\ - 1 & 4 \end{array} \right)\), where \(a\) is a constant.

1. Find the value of \(a\) for which the matrix \(\mathbf { A }\) is singular. $$\mathbf { B } = \left( \begin{array} { r r } 3 & - 2 \\ - 1 & 4 \end{array} \right)$$
2. Find \(\mathbf { B } ^ { - 1 }\). The transformation represented by \(\mathbf { B }\) maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( k - 6,3 k + 12 )\), where \(k\) is a constant,
3. show that \(P\) lies on the line with equation \(y = x + 3\).

2. \(\mathbf { M } = \left( \begin{array} { c c } 2 a & 3 \\ 6 & a \end{array} \right)\), where \(a\) is a real constant.

1. Given that \(a = 2\), find \(\mathbf { M } ^ { - 1 }\).
2. Find the values of \(a\) for which \(\mathbf { M }\) is singular.

$$\mathbf { A } = \left( \begin{array} { r r } 5 k & 3 k - 1 \\ - 3 & k + 1 \end{array} \right) , \text { where } k \text { is a real constant. }$$ Given that \(\mathbf { A }\) is a singular matrix, find the possible values of \(k\).
(ii) $$\mathbf { B } = \left( \begin{array} { l l } 10 & 5 \\ - 3 & 3 \end{array} \right)$$ A triangle \(T\) is transformed onto a triangle \(T ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { B }\). The vertices of triangle \(T ^ { \prime }\) have coordinates \(( 0,0 ) , ( - 20,6 )\) and \(( 10 c , 6 c )\), where \(c\) is a positive constant. The area of triangle \(T ^ { \prime }\) is 135 square units.

1. Find the matrix \(\mathbf { B } ^ { - 1 }\)
2. Find the coordinates of the vertices of the triangle \(T\), in terms of \(c\) where necessary.
3. Find the value of \(c\).

2. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 1 \\ 4 & 3 \end{array} \right) , \quad \mathbf { P } = \left( \begin{array} { r r } 3 & 6 \\ 11 & - 8 \end{array} \right)$$

1. Find \(\mathbf { A } ^ { - 1 }\) (2) The transformation represented by the matrix \(\mathbf { B }\) followed by the transformation represented by the matrix \(\mathbf { A }\) is equivalent to the transformation represented by the matrix \(\mathbf { P }\).
2. Find \(\mathbf { B }\), giving your answer in its simplest form.

3. $$\mathbf { A } = \left( \begin{array} { l l l } 2 & k & 2 \\ 2 & 2 & k \\ 1 & 2 & 2 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Determine the values of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
2. find \(\mathbf { A } ^ { - 1 }\), giving your answer in terms of \(k\).
   3.

2. $$\mathbf { T } = \left( \begin{array} { l l l } 2 & 3 & 7 \\ 3 & 2 & 6 \\ a & 4 & b \end{array} \right) \quad \mathbf { U } = \left( \begin{array} { r r r } 6 & - 1 & - 4 \\ 15 & c & - 9 \\ - 8 & a & 5 \end{array} \right)$$ where \(a\), \(b\) and \(c\) are constants.
Given that \(\mathbf { T U } = \mathbf { I }\)

1. determine the value of \(a\), the value of \(b\) and the value of \(c\) The transformation represented by the matrix \(\mathbf { T }\) transforms the line \(l _ { 1 }\) to the line \(l _ { 2 }\) Given that \(l _ { 2 }\) has equation $$\frac { x - 1 } { 3 } = \frac { y } { - 4 } = z + 2$$
2. determine a Cartesian equation for \(l _ { 1 }\)

4. A non-singular matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { l l l } 3 & k & 0 \\ k & 2 & 0 \\ k & 0 & 1 \end{array} \right) \text {, where } k \text { is a constant. }$$

1. Find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\). The point \(A\) is mapped onto the point ( \(- 5,10,7\) ) by the transformation represented by the matrix $$\left( \begin{array} { l l l } 3 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 0 & 1 \end{array} \right)$$
2. Find the coordinates of the point \(A\).

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

1. Find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). Hence, given that \(k = 0\)
2. find the matrix \(\mathbf { N }\) such that $$\mathbf { M N } = \left( \begin{array} { r r r } 3 & 5 & 6 \\ 4 & - 1 & 1 \\ 3 & 2 & - 3 \end{array} \right)$$

5. $$\mathbf { M } = \left( \begin{array} { r r r } 4 & - 5 & 0 \\ k & 2 & 0 \\ - 3 & - 5 & k \end{array} \right) \text {, where } k \text { is a real constant, } k \neq 0 , k \neq - \frac { 8 } { 5 }$$

1. Find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\). A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix $$\left( \begin{array} { r r r } 4 & - 5 & 0 \\ - 1 & 2 & 0 \\ - 3 & - 5 & - 1 \end{array} \right)$$ The transformation \(T\) maps the plane \(\Pi _ { 1 }\) onto the plane \(\Pi _ { 2 }\) Given that the plane \(\Pi _ { 2 }\) has equation \(2 x - z = 4\)
2. find a cartesian equation of the plane \(\Pi _ { 1 }\)

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 \(l _ { 1 }\)

3. $$\mathbf { M } = \left( \begin{array} { r r r } - 2 & 5 & 0 \\ 5 & 1 & - 3 \\ 0 & - 3 & 6 \end{array} \right)$$ Given that \(\mathbf { i } + \mathbf { j } + \mathbf { k }\) is an eigenvector of \(\mathbf { M }\),

1. determine the corresponding eigenvalue. Given that 8 is an eigenvalue of \(\mathbf { M }\),
2. determine a corresponding eigenvector.
3. Determine a diagonal matrix \(\mathbf { D }\) and an orthogonal matrix \(\mathbf { P }\) such that $$\mathbf { D } = \mathbf { P } ^ { \mathrm { T } } \mathbf { M P }$$

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 \(x , \mathbf { A } ^ { - 1 }\)

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 0 & 0 \\ 0 & 1 & 4 \\ 3 & - 2 & - 3 \end{array} \right)$$

1. Determine \(\mathbf { M } ^ { - 1 }\) The transformation represented by \(\mathbf { M }\) maps the plane \(\Pi _ { 1 }\) to the plane \(\Pi _ { 2 }\) The point \(( x , y , z )\) on \(\Pi _ { 1 }\) maps to the point \(( u , v , w )\) on \(\Pi _ { 2 }\)
2. Determine \(x , y\) and \(z\) in terms of \(u , v\) and \(w\) as appropriate. The plane \(\Pi _ { 1 }\) has equation $$3 x - 7 y + 2 z = - 3$$
3. Find a Cartesian equation for \(\Pi _ { 2 }\) Give your answer in the form \(a u + b v + c w = d\) where \(a , b , c\) and \(d\) are integers to be determined.

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.

4. The matrix \(\mathbf { M }\) is given by $$\left( \begin{array} { r r r } 2 & 0 & - 1 \\ k & 3 & 2 \\ - 2 & 1 & k \end{array} \right)$$

1. Show that \(\operatorname { det } \mathbf { M } = 5 k - 10\) Given that \(k \neq 2\)
2. find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). The points \(O ( 0,0,0 ) , A ( 4 , - 8,3 ) , B ( - 2,5 , - 4 )\) and \(C ( 4 , - 6,8 )\) are the vertices of a tetrahedron \(T\). The transformation represented by matrix \(\mathbf { M }\) transforms \(T\) to a tetrahedron with volume 50
3. Determine the possible values of \(k\).

6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3 \\ 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant. Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\),

1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\),
2. show that \(k = 3\),
3. show that \(\mathbf { M }\) has exactly two eigenvalues. A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\).
   The transformation \(T\) maps the line \(l _ { 1 }\), with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\), onto the line \(l _ { 2 }\).
4. Taking \(k = 3\), find cartesian equations of \(l _ { 2 }\).

1. The matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } 1 & 1 & a \\ 2 & b & c \\ - 1 & 0 & 1 \end{array} \right) , \text { where } a , b \text { and } c \text { are constants. }$$

1. Given that \(\mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - \mathbf { k }\) are two of the eigenvectors of \(\mathbf { M }\), find
   1. the values of \(a , b\) and \(c\),
   2. the eigenvalues which correspond to the two given eigenvectors.
2. The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 1 & 0 \\ 2 & 1 & d \\ - 1 & 0 & 1 \end{array} \right) \text {, where } d \text { is constant, } d \neq - 1$$ Find
   1. the determinant of \(\mathbf { P }\) in terms of \(d\),
   2. the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(d\).

6. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & 0 \\ 2 & - 2 & 1 \\ - 4 & 1 & - 1 \end{array} \right) , k \in \mathbb { R } , k \neq \frac { 1 } { 2 }$$

1. Show that \(\operatorname { det } \mathbf { M } = 1 - 2 k\).
2. Find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix $$\left( \begin{array} { r r r } 1 & 0 & 0 \\ 2 & - 2 & 1 \\ - 4 & 1 & - 1 \end{array} \right)$$ Given that \(l _ { 2 }\) has cartesian equation $$\frac { x - 1 } { 5 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 1 }$$
3. find a cartesian equation of the line \(l _ { 1 }\)

3

1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 4 & 1 & k \\ 3 & 2 & 5 \\ 8 & 5 & 13 \end{array} \right)\), where \(k \neq 5\).
2. Solve the simultaneous equations $$\begin{aligned} & 4 x + y + 7 z = 12 \\ & 3 x + 2 y + 5 z = m \\ & 8 x + 5 y + 13 z = 0 \end{aligned}$$ giving \(x , y\) and \(z\) in terms of \(m\).
3. Find the value of \(p\) for which the simultaneous equations $$\begin{aligned} & 4 x + y + 5 z = 12 \\ & 3 x + 2 y + 5 z = p \\ & 8 x + 5 y + 13 z = 0 \end{aligned}$$ have solutions, and find the general solution in this case.