4.03o Inverse 3x3 matrix

193 questions

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CAIE FP1 2019 November Q8
10 marks Standard +0.8
The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{pmatrix},$$ where \(m \neq 0, 1, 2\).
  1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M} = \mathbf{PDP}^{-1}\). [7]
  2. Find \(\mathbf{M}^T\mathbf{P}\). [3]
CAIE Further Paper 2 2020 June Q3
8 marks Challenging +1.2
The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{pmatrix} 5 & -1 & 7 \\ 0 & 6 & 0 \\ 7 & 7 & 5 \end{pmatrix}.$$
  1. Find the eigenvalues of \(\mathbf{A}\). [4]
  2. Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\). [4]
CAIE Further Paper 2 2021 November Q6
11 marks Challenging +1.2
The matrix \(\mathbf{P}\) is given by $$\mathbf{P} = \begin{pmatrix} 1 & 6 & 6 \\ 0 & 2 & 6 \\ 0 & 0 & -3 \end{pmatrix}.$$
  1. Use the characteristic equation of \(\mathbf{P}\) to find \(\mathbf{P}^{-1}\). [5]
  2. Find the matrix \(\mathbf{A}\) such that $$\mathbf{P}^{-1}\mathbf{A}\mathbf{P} = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{pmatrix}.$$ [4]
  3. State the eigenvalues and corresponding eigenvectors of \(\mathbf{A}^3\). [2]
CAIE Further Paper 2 2023 November Q7
11 marks Challenging +1.2
The matrix A is given by $$\mathbf{A} = \begin{pmatrix} -6 & 2 & 13 \\ 0 & -2 & 5 \\ 0 & 0 & 8 \end{pmatrix}.$$
  1. Find a matrix P and a diagonal matrix D such that \(\mathbf{A}^{-1} = \mathbf{PDP}^{-1}\). [7]
  2. Use the characteristic equation of A to find \(\mathbf{A}^{-1}\). [4]
Edexcel F3 2021 June Q3
9 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} 3 & 1 & p \\ 1 & 1 & 2 \\ -1 & p & 2 \end{pmatrix}$$ where \(p\) is a real constant
  1. Find the exact values of \(p\) for which \(\mathbf{M}\) has no inverse. [4]
Given that \(\mathbf{M}\) does have an inverse,
  1. find \(\mathbf{M}^{-1}\) in terms of \(p\). [5]
Edexcel F3 2018 Specimen Q4
9 marks Standard +0.3
$$\mathbf{M} = \begin{pmatrix} 1 & k & 0 \\ -1 & 1 & 1 \\ 1 & k & 3 \end{pmatrix}, \text{ where } k \text{ is a constant}$$
  1. Find \(\mathbf{M}^{-1}\) in terms of \(k\). [5]
Hence, given that \(k = 0\)
  1. find the matrix \(\mathbf{N}\) such that $$\mathbf{MN} = \begin{pmatrix} 3 & 5 & 6 \\ 4 & -1 & 1 \\ 3 & 2 & -3 \end{pmatrix}$$ [4]
Edexcel FP3 2011 June Q7
12 marks Challenging +1.2
The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} k & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}, \quad k \neq 1$$
  1. Show that \(\det \mathbf{M} = 2 - 2k\). [2]
  2. Find \(\mathbf{M}^{-1}\), in terms of \(k\). [5] The straight line \(l_1\) is mapped onto the straight line \(l_2\) by the transformation represented by the matrix \(\begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}\). The equation of \(l_2\) is \((\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}\), where \(\mathbf{a} = 4\mathbf{i} + \mathbf{j} + 7\mathbf{k}\) and \(\mathbf{b} = 4\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).
  3. Find a vector equation for the line \(l_1\). [5]
Edexcel FP3 Q19
11 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 5 & 3 & u \end{pmatrix}, \quad u \neq 1.$$
  1. Show that \(\det \mathbf{A} = 2(u - 1)\). [2]
  2. Find the inverse of \(\mathbf{A}\). [6]
The image of the vector \(\begin{pmatrix} a \\ b \\ c \end{pmatrix}\) when transformed by the matrix \(\begin{pmatrix} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 5 & 3 & 6 \end{pmatrix}\) is \(\begin{pmatrix} 3 \\ 1 \\ 6 \end{pmatrix}\).
  1. Find the values of \(a\), \(b\) and \(c\). [3]
Edexcel FP3 Q35
9 marks Challenging +1.3
    1. Explain why, for any two vectors \(\mathbf{a}\) and \(\mathbf{b}\), \(\mathbf{a} \cdot \mathbf{b} \times \mathbf{a} = 0\). [2]
    2. Given vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) such that \(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}\), where \(\mathbf{a} \neq \mathbf{0}\) and \(\mathbf{b} \neq \mathbf{c}\), show that $$\mathbf{b} - \mathbf{c} = \lambda\mathbf{a}, \quad \text{where } \lambda \text{ is a scalar.}$$ [2]
  2. \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are \(2 \times 2\) matrices.
    1. Given that \(\mathbf{A}\mathbf{B} = \mathbf{A}\mathbf{C}\), and that \(\mathbf{A}\) is not singular, prove that \(\mathbf{B} = \mathbf{C}\). [2]
    2. Given that \(\mathbf{A}\mathbf{B} = \mathbf{A}\mathbf{C}\), where \(\mathbf{A} = \begin{pmatrix} 3 & 6 \\ 1 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix}\), find a matrix \(\mathbf{C}\) whose elements are all non-zero. [3]
Edexcel FP3 Q37
14 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & k \end{pmatrix}.$$
  1. Show that \(\det \mathbf{A} = 20 - 4k\). [2]
  2. Find \(\mathbf{A}^{-1}\). [6]
Given that \(k = 3\) and that \(\begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\),
  1. find the corresponding eigenvalue. [2]
Given that the only other distinct eigenvalue of \(\mathbf{A}\) is \(8\),
  1. find a corresponding eigenvector. [4]
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)
AQA FP1 2016 June Q8
10 marks Standard +0.3
The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\).
    1. Find the matrix \(\mathbf{A}^2\). [1 mark]
    2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf{A}^2\). [1 mark]
  2. Given that the matrix \(\mathbf{B}\) represents a reflection in the line \(x + \sqrt{3}y = 0\), find the matrix \(\mathbf{B}\), giving the exact values of any trigonometric expressions. [2 marks]
  3. Hence find the coordinates of the point \(P\) which is mapped onto \((0, -4)\) under the transformation represented by \(\mathbf{A}^2\) followed by a reflection in the line \(x + \sqrt{3}y = 0\). [6 marks]
OCR FP1 Q2
6 marks Standard +0.3
The matrices \(\mathbf{A}\) and \(\mathbf{I}\) are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}\) and \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) respectively.
  1. Find \(\mathbf{A}^2\) and verify that \(\mathbf{A}^2 = 4\mathbf{A} - \mathbf{I}\). [4]
  2. Hence, or otherwise, show that \(\mathbf{A}^{-1} = 4\mathbf{I} - \mathbf{A}\). [2]
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]
OCR MEI FP1 2006 June Q5
6 marks Moderate -0.3
  1. The matrix \(\mathbf{S} = \begin{pmatrix} -1 & 2 \\ -3 & 4 \end{pmatrix}\) represents a transformation.
    1. Show that the point \((1, 1)\) is invariant under this transformation. [1]
    2. Calculate \(\mathbf{S}^{-1}\). [2]
    3. Verify that \((1, 1)\) is also invariant under the transformation represented by \(\mathbf{S}^{-1}\). [1]
  2. Part (i) may be generalised as follows. If \((x, y)\) is an invariant point under a transformation represented by the non-singular matrix \(\mathbf{T}\), it is also invariant under the transformation represented by \(\mathbf{T}^{-1}\). Starting with \(\mathbf{T}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\), or otherwise, prove this result. [2]
OCR MEI FP1 2007 June Q1
3 marks Moderate -0.8
You are given the matrix \(\mathbf{M} = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}\).
  1. Find the inverse of \(\mathbf{M}\). [2]
  2. A triangle of area 2 square units undergoes the transformation represented by the matrix \(\mathbf{M}\). Find the area of the image of the triangle following this transformation. [1]
OCR MEI FP1 2007 June Q10
11 marks Standard +0.8
You are given that \(\mathbf{A} = \begin{pmatrix} 1 & -2 & k \\ 2 & 1 & 2 \\ 3 & 2 & -1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} -5 & -2+2k & -4-k \\ 8 & -1-3k & -2+2k \\ 1 & -8 & 5 \end{pmatrix}\) and that \(\mathbf{AB}\) is of the form \(\mathbf{AB} = \begin{pmatrix} k-n & 0 & 0 \\ 0 & k-n & 0 \\ 0 & 0 & k-n \end{pmatrix}\).
  1. Find the value of \(n\). [2]
  2. Write down the inverse matrix \(\mathbf{A}^{-1}\) and state the condition on \(k\) for this inverse to exist. [4]
  3. Using the result from part (ii), or otherwise, solve the following simultaneous equations. \begin{align} x - 2y + z &= 1
    2x + y + 2z &= 12
    3x + 2y - z &= 3 \end{align} [5]
OCR MEI FP2 2009 June Q2
19 marks Standard +0.3
  1. Obtain the characteristic equation for the matrix \(\mathbf{M}\) where $$\mathbf{M} = \begin{pmatrix} 3 & 1 & -2 \\ 6 & -1 & 0 \\ 2 & 0 & 1 \end{pmatrix}.$$ Hence or otherwise obtain the value of \(\det(\mathbf{M})\). [3]
  2. Show that \(-1\) is an eigenvalue of \(\mathbf{M}\), and show that the other two eigenvalues are not real. Find an eigenvector corresponding to the eigenvalue \(-1\). Hence or otherwise write down the solution to the following system of equations. [9] $$3x + y - 2z = -0.1$$ $$-y = 0.6$$ $$2x + z = 0.1$$
  3. State the Cayley-Hamilton theorem and use it to show that $$\mathbf{M}^3 = 3\mathbf{M}^2 - 3\mathbf{M} - 7\mathbf{I}.$$ Obtain an expression for \(\mathbf{M}^{-1}\) in terms of \(\mathbf{M}^2\), \(\mathbf{M}\) and \(\mathbf{I}\). [4]
  4. Find the numerical values of the elements of \(\mathbf{M}^{-1}\), showing your working. [3]
AQA Further AS Paper 1 2018 June Q16
3 marks Standard +0.8
Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = \mathbf{I} + 2\mathbf{A}$$ where \(\mathbf{I}\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]
AQA Further AS Paper 1 2020 June Q16
4 marks Moderate -0.8
\(\mathbf{A}\) and \(\mathbf{B}\) are non-singular square matrices.
  1. Write down the product \(\mathbf{AA}^{-1}\) as a single matrix. [1 mark]
  2. \(\mathbf{M}\) is a matrix such that \(\mathbf{M} = \mathbf{AB}\). Prove that \(\mathbf{M}^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\) [3 marks]
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 Q6
11 marks Challenging +1.2
The matrix M is given by $$\mathbf{M} = \frac{1}{10} \begin{pmatrix} a & a & -6 \\ 0 & 10 & 0 \\ 9 & 14 & -13 \end{pmatrix}$$ where \(a\) is a real number. The vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) are eigenvectors of \(\mathbf{M}\) The corresponding eigenvalues are \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\) respectively. It is given that \(\lambda_2 = 1\) and \(\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}\), \(\mathbf{v}_2 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\) and \(\mathbf{v}_3 = \begin{pmatrix} c \\ 0 \\ 1 \end{pmatrix}\), where \(c\) is an integer.
    1. Find the value of \(\lambda_1\) [2 marks]
    2. Find the value of \(a\) [2 marks]
  2. Find the integer \(c\) and the value of \(\lambda_3\) [4 marks]
  3. Find matrices \(\mathbf{U}\), \(\mathbf{D}\) and \(\mathbf{U}^{-1}\), such that \(\mathbf{D}\) is diagonal and \(\mathbf{M} = \mathbf{UDU}^{-1}\) [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]