Find inverse then solve system

Questions requiring the student to first compute the inverse of a 3×3 matrix (possibly in terms of a parameter) and then use it to solve a system of three linear equations.

25 questions · Standard +0.4

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Edexcel F3 2014 June Q4
8 marks Standard +0.3
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 \(A\).
OCR FP1 2007 January Q10
11 marks Standard +0.3
10 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { r r r } a & 2 & 0 \\ 3 & 1 & 2 \\ 0 & - 1 & 1 \end{array} \right)\), where \(a \neq 2\).
  1. Find \(\mathbf { D } ^ { - 1 }\).
  2. Hence, or otherwise, solve the equations $$\begin{aligned} a x + 2 y & = 3 \\ 3 x + y + 2 z & = 4 \\ - y + z & = 1 \end{aligned}$$
OCR FP1 2013 June Q10
12 marks Standard +0.3
10 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 1 \\ 1 & 3 & 2 \\ 4 & 1 & 1 \end{array} \right)\).
  1. Find the value of \(a\) for which \(\mathbf { A }\) is singular.
  2. Given that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the equations $$\begin{aligned} a x + 2 y + z & = 1 \\ x + 3 y + 2 z & = 2 \\ 4 x + y + z & = 3 \end{aligned}$$
OCR FP1 Specimen Q8
14 marks Standard +0.8
8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 1 \\ 2 & 3 & - 1 \\ 2 & - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
  1. Show that the determinant of \(\mathbf { M }\) is \(2 a\).
  2. Given that \(a \neq 0\), find the inverse matrix \(\mathbf { M } ^ { - 1 }\).
  3. Hence or otherwise solve the simultaneous equations $$\begin{array} { r } x + 2 y - z = 1 \\ 2 x + 3 y - z = 2 \\ 2 x - y + z = 0 \end{array}$$
  4. Find the value of \(k\) for which the simultaneous equations $$\begin{array} { r } 2 y - z = k \\ 2 x + 3 y - z = 2 \\ 2 x - y + z = 0 \end{array}$$ have solutions.
  5. Do the equations in part (iv), with the value of \(k\) found, have a solution for which \(x = z\) ? Justify your answer.
OCR FP1 2010 January Q9
11 marks Standard +0.3
9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 0 & 3 & 1 \\ 1 & 1 & a \end{array} \right)\), where \(a \neq 1\).
  1. Find \(\mathbf { A } ^ { - 1 }\).
  2. Hence, or otherwise, solve the equations $$\begin{array} { r } 2 x - y + z = 1 \\ 3 y + z = 2 \\ x + y + a z = 2 \end{array}$$
OCR Further Pure Core AS 2024 June Q1
4 marks Standard +0.3
1 Use a matrix method to determine the solution of the following simultaneous equations. $$\begin{aligned} 2 x - 3 y + z & = 1 \\ x - 2 y - 4 z & = 40 \\ 5 x + 6 y - z & = 61 \end{aligned}$$
OCR Further Pure Core 2 2022 June Q7
13 marks Standard +0.8
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 \(\mathbf { A } \left( \begin{array} { c } \mathrm { x } \\ \mathrm { y } \\ \mathrm { z } \end{array} \right) = \left( \begin{array} { r } 6 \\ - 9 \\ 11 \end{array} \right)\).
    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.
    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.
OCR MEI Further Pure Core AS 2019 June Q4
8 marks Standard +0.3
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 \(- x + y + 2 z = 4\).
Edexcel CP AS 2018 June Q1
5 marks Standard +0.3
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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Edexcel CP2 2024 June Q8
7 marks Standard +0.3
8. $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 1 \\ 1 & 1 & 1 \\ k & 3 & 6 \end{array} \right) \quad k \neq 0$$
  1. Find, in terms of \(k , \mathbf { A } ^ { - 1 }\)
  2. Determine, in simplest form in terms of \(k\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 3 x + y - z = 3 \\ x + y + z = 1 \\ k x + 3 y + 6 z = 6 \end{array}$$
Pre-U Pre-U 9795/1 2010 June Q3
4 marks Moderate -0.8
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 \(r\).
Pre-U Pre-U 9795 Specimen Q9
Standard +0.8
9
  1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 1 & 3 & 4 \\ 2 & 5 & - 1 \\ 3 & 8 & 2 \end{array} \right)\), and hence solve the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 10 \\ 3 x + 8 y + 2 z & = 8 \end{aligned}$$
  2. Find the value of \(k\) for which the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 15 \\ 3 x + 8 y + 3 z & = k \end{aligned}$$ is consistent. Find the solution in this case and interpret it geometrically.
Pre-U Pre-U 9795/1 Specimen Q8
10 marks Standard +0.8
8
  1. Show that if \(a \neq 3\) then the system of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 5 a \\ 3 x + 8 y + a z & = b \end{aligned}$$ has a unique solution.
  2. By use of the inverse matrix of a suitable \(3 \times 3\) matrix, find the unique solution in the case \(a = 1\) and \(b = 2\).
  3. Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.
Edexcel FP3 Specimen Q7
12 marks Standard +0.3
$$\mathbf{A}(x) = \begin{pmatrix} 1 & x & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}, \quad x \neq \frac{5}{2}$$
  1. Calculate the inverse of \(\mathbf{A}(x)\). $$\mathbf{B} = \begin{pmatrix} 1 & 3 & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}$$ [8] The image of the vector \(\begin{pmatrix} p \\ q \\ r \end{pmatrix}\) when transformed by \(\mathbf{B}\) is \(\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}\)
  2. Find the values of \(p\), \(q\) and \(r\). [4]
(Total 14 marks)
OCR FP1 Q7
10 marks Standard +0.3
The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} a & 1 & 3 \\ 2 & 1 & -1 \\ 0 & 1 & 2 \end{pmatrix}\).
  1. Given that \(\mathbf{B}\) is singular, show that \(a = -\frac{2}{3}\). [3]
  2. Given instead that \(\mathbf{B}\) is non-singular, find the inverse matrix \(\mathbf{B}^{-1}\). [4]
  3. Hence, or otherwise, solve the equations \begin{align} -x + y + 3z &= 1,
    2x + y - z &= 4,
    y + 2z &= -1. \end{align} [3]
OCR FP1 2005 June Q7
10 marks Standard +0.3
The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} a & 1 & 3 \\ 2 & 1 & -1 \\ 0 & 1 & 2 \end{pmatrix}\).
  1. Given that \(\mathbf{B}\) is singular, show that \(a = -\frac{2}{3}\). [3]
  2. Given instead that \(\mathbf{B}\) is non-singular, find the inverse matrix \(\mathbf{B}^{-1}\). [4]
  3. Hence, or otherwise, solve the equations \begin{align} -x + y + 3z &= 1,
    2x + y - z &= 4,
    y + 2z &= -1. \end{align} [3]
AQA Further Paper 1 2021 June Q12
14 marks Standard +0.8
The matrix \(\mathbf{A} = \begin{pmatrix} 1 & 5 & 3 \\ 4 & -2 & p \\ 8 & 5 & -11 \end{pmatrix}\), where \(p\) is a constant.
  1. Given that A is a non-singular matrix, find \(\mathbf{A}^{-1}\) in terms of \(p\). State any restrictions on the value of \(p\). [6 marks]
  2. The equations below represent three planes. \(x + 5y + 3z = 5\) \(4x - 2y + pz = 24\) \(8x + 5y - 11z = -30\)
    1. Find, in terms of \(p\), the coordinates of the point of intersection of the three planes. [4 marks]
    2. In the case where \(p = 2\), show that the planes are mutually perpendicular. [4 marks]
AQA Further Paper 1 2022 June Q7
9 marks Standard +0.3
The matrix \(\mathbf{M}\) is defined as $$\mathbf{M} = \begin{bmatrix} 1 & 7 & -3 \\ 3 & 6 & k+1 \\ 1 & 3 & 2 \end{bmatrix}$$ where \(k\) is a constant.
    1. Given that \(\mathbf{M}\) is a non-singular matrix, find \(\mathbf{M}^{-1}\) in terms of \(k\) [5 marks]
    2. State any restrictions on the value of \(k\) [1 mark]
  2. Using your answer to part (a)(i), solve \begin{align} x + 7y - 3z &= 6
    3x + 6y + 6z &= 3
    x + 3y + 2z &= 1 \end{align} [3 marks]
AQA Further Paper 1 2023 June Q10
12 marks Challenging +1.2
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]
AQA Further Paper 2 2024 June Q14
10 marks Standard +0.8
The matrix \(\mathbf{M}\) is defined as $$\mathbf{M} = \begin{bmatrix} 5 & 2 & 1 \\ 6 & 3 & 2k + 3 \\ 2 & 1 & 5 \end{bmatrix}$$ where \(k\) is a constant.
  1. Given that \(\mathbf{M}\) is a non-singular matrix, find \(\mathbf{M}^{-1}\) in terms of \(k\) [5 marks]
  2. State any restrictions on the value of \(k\) [1 mark]
  3. Using your answer to part (a), show that the solution to the set of simultaneous equations below is independent of the value of \(k\) \(5x + 2y + z = 1\) \(6x + 3y + (2k + 3)z = 4k + 3\) \(2x + y + 5z = 9\) [4 marks]
OCR Further Pure Core 2 2024 June Q3
7 marks Standard +0.3
Matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 4 & -3 \\ -2 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 3 & -5 \\ 0 & 1 \end{pmatrix}\).
  1. Find \(2\mathbf{A} - 4\mathbf{B}\). [2]
  2. Write down the matrix \(\mathbf{C}\) such that \(\mathbf{A}\mathbf{C} = 2\mathbf{A}\). [1]
  3. Find the value of \(\det \mathbf{A}\). [1]
  4. In this question you must show detailed reasoning. Use \(\mathbf{A}^{-1}\) to solve the equations \(4x - 3y = 7\) and \(-2x + 2y = 9\). [3]
WJEC Further Unit 4 2019 June Q3
8 marks Standard +0.3
  1. Determine whether or not the following set of equations $$\begin{pmatrix} 2 & -7 & 2 \\ 0 & 3 & -2 \\ -7 & 8 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$ has a unique solution, where \(a\), \(b\), \(c\) are constants. [3]
  2. Solve the set of equations \begin{align} x + 8y - 6z &= 5,
    2x + 4y + 6z &= -3,
    -5x - 4y + 9z &= -7. \end{align} Show all your working. [5]
WJEC Further Unit 4 2023 June Q4
5 marks Moderate -0.3
Solve the simultaneous equations \begin{align} 4x - 2y + 3z &= 8,
2x - 3y + 8z &= -1,
2x + 4y - z &= 0. \end{align} [5]
WJEC Further Unit 4 Specimen Q6
7 marks Standard +0.3
The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 3 & 2 \\ 3 & 2 & 5 \end{pmatrix}.$$
  1. Find
    1. the adjugate matrix of \(\mathbf{M}\),
    2. hence determine the inverse matrix \(\mathbf{M}^{-1}\). [5]
  2. Use your result to solve the simultaneous equations \begin{align} 2x + y + 3z &= 13
    x + 3y + 2z &= 13
    3x + 2y + 5z &= 22 \end{align} [2]
Pre-U Pre-U 9795 Specimen Q10
10 marks Standard +0.3
  1. Find the inverse of the matrix \(\begin{pmatrix} 1 & 3 & 4 \\ 2 & 5 & -1 \\ 3 & 8 & 2 \end{pmatrix}\), and hence solve the set of equations \begin{align} x + 3y + 4z &= -5,
    2x + 5y - z &= 10,
    3x + 8y + 2z &= 8. \end{align} [5]
  2. Find the value of \(k\) for which the set of equations \begin{align} x + 3y + 4z &= -5,
    2x + 5y - z &= 15,
    3x + 8y + 3z &= k, \end{align} is consistent. Find the solution in this case and interpret it geometrically. [5]