4.03r Solve simultaneous equations: using inverse matrix

$$\mathbf{A} = \begin{pmatrix} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 5 & 3 & u \end{pmatrix}, \quad u \neq 1.$$

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]

$$\mathbf{A}(x) = \begin{pmatrix} 1 & x & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}, \quad x \neq \frac{5}{2}$$

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)

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]

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]

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]

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]

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. 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]

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]

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]

You are given the system of equations $$a^2x - 2y = 1$$ $$x + b^2y = 3$$ where \(a\) and \(b\) are real numbers.

1. Use a matrix method to find \(x\) and \(y\) in terms of \(a\) and \(b\). [4]
2. Explain why the method used in part (a) works for all values of \(a\) and \(b\). [2]

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]

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that \(\mathbf{A} = \begin{bmatrix} 4 & 2 \\ -1 & -3 \end{bmatrix}\) and \(\mathbf{B} = \begin{bmatrix} 4 & 2 \\ 2 & 1 \end{bmatrix}\).

1. Explain why \(\mathbf{B}\) has no inverse. [1]
2. 1. Find the inverse of \(\mathbf{A}\). [3]
   2. Hence, find the matrix \(\mathbf{X}\), where \(\mathbf{AX} = \begin{bmatrix} -4 \\ 1 \end{bmatrix}\) [2]

Solve the simultaneous equations \begin{align} 4x - 2y + 3z &= 8,
2x - 3y + 8z &= -1,
2x + 4y - z &= 0. \end{align} [5]

The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 12 & 30 & 8 \\ 18 & 25 & 20 \\ 19 & 50 & 16 \end{pmatrix}.$$

1. Given that \(\det \mathbf{M} = -1040\), give a geometrical interpretation of the solution to the following equation. [2] $$\begin{pmatrix} 12 & 30 & 8 \\ 18 & 25 & 20 \\ 19 & 50 & 16 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2668 \\ 3402 \\ 4581 \end{pmatrix}$$
2. Three hotels A, B, C each have different types of room available to book: single, double and family rooms. For each type of room, the price per night is the same in each of the three hotels. The table below gives, for each hotel, details of the number of each type of room and the total revenue per night when the hotel is full.

   \multirow{2}{*}{Hotel}	Types of room			\multirow{2}{*}{Total revenue}
   \cline{2-4}	Single	Double	Family
   A	12	30	8	£2,668
   B	18	25	20	£3,402
   C	19	50	16	£4,581

   Find the price per night of each type of room. [6]

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]

\(\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}\), where \(k\) is a constant.

1. Show that the matrix \(\mathbf{A}\) is non-singular for all values of \(k\). [2]

A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\). The point \(P\) has position vector \(\begin{pmatrix} a \\ 2a \end{pmatrix}\) relative to an origin \(O\). The point \(Q\) has position vector \(\begin{pmatrix} 7 \\ -3 \end{pmatrix}\) relative to \(O\). Given that the point \(P\) is mapped onto the point \(Q\) under \(T\),

1. determine the value of \(a\) and the value of \(k\). [3]

Given that, for a different value of \(k\), \(T\) maps the line \(y = 2x\) onto itself,

1. determine this value of \(k\). [3]

$$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ 3 & k & 4 \\ 3 & 2 & -1 \end{pmatrix} \quad \text{where } k \text{ is a constant}$$

1. Find the values of \(k\) for which the matrix \(\mathbf{M}\) has an inverse. [2]
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect \begin{align} 2x - y + z &= p
   3x - 6y + 4z &= 1
   3x + 2y - z &= 0 \end{align} [5]
3. 1. Find the value of \(q\) for which the set of simultaneous equations \begin{align} 2x - y + z &= 1
      3x - 5y + 4z &= q
      3x + 2y - z &= 0 \end{align} can be solved.
   2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically. [4]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & a \\ 0 & 1 \end{pmatrix}\), where \(a\) is a constant.

1. Find \(\mathbf{A}^{-1}\). [2]

The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} 2 & a \\ 4 & 1 \end{pmatrix}\).

1. Given that \(\mathbf{PA} = \mathbf{B}\), find the matrix \(\mathbf{P}\). [3]

You are given that \(\mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & -2 & -1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 1 & 0 & 1 \\ -8 & 4 & 0 \\ 19 & -8 & -1 \end{pmatrix}\).

1. Find \(\mathbf{AB}\). [1]
2. Hence write down \(\mathbf{A}^{-1}\). [1]
3. You are given three simultaneous equations $$x + 2y + z = 0$$ $$2x + 5y + 2z = 1$$ $$3x - 2y - z = 4$$
   1. Explain how you can tell, without solving them, that there is a unique solution to these equations. [2]
   2. Find this unique solution. [2]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} -3 & 3 & 2 \\ 5 & -4 & -3 \\ -1 & 1 & 1 \end{pmatrix}\).

1. Find \(\mathbf{A}^{-1}\). [1]
2. Solve the simultaneous equations $$-3x + 3y + 2z = 12a$$ $$5x - 4y - 3z = -6$$ $$-x + y + z = 7$$ giving your solution in terms of \(a\). [3]

In the following set of simultaneous equations, \(a\) and \(b\) are constants. \begin{align} 3x + 2y - z &= 5
2x - 4y + 7z &= 60
ax + 20y - 25z &= b \end{align}

1. In the case where \(a = 10\), solve the simultaneous equations, giving your solution in terms of \(b\). [3]
2. Determine the value of \(a\) for which there is no unique solution for \(x\), \(y\) and \(z\). [3]
3. 1. Find the values of \(\alpha\) and \(\beta\) for which \(\alpha(2y - z) + \beta(-4y + 7z) = 20y - 25z\) for any \(y\) and \(z\). [3]
   2. Hence, for the case where there is no unique solution for \(x\), \(y\) and \(z\), determine the value of \(b\) for which there is an infinite number of solutions. [2]
   3. When \(a\) takes the value in part (ii) and \(b\) takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations. [1]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & 2 & 3 \\ 4 & 4 & 6 \\ -2 & 2 & 9 \end{pmatrix}\) where \(a\) is a constant. It is given that if \(\mathbf{A}\) is not singular then $$\mathbf{A}^{-1} = \frac{1}{24a-48} \begin{pmatrix} 24 & -12 & 0 \\ -48 & 9a+6 & 12-6a \\ 16 & -2a-4 & 4a-8 \end{pmatrix}.$$

1. Use \(\mathbf{A}^{-1}\) to solve the simultaneous equations below, giving your answer in terms of \(k\). \begin{align} x + 2y + 3z &= 6
   4x + 4y + 6z &= 8
   -2x + 2y + 9z &= k \end{align} [3]
2. Consider the equations below where \(a\) takes the value which makes \(\mathbf{A}\) singular. \begin{align} ax + 2y + 3z &= b
   4x + 4y + 6z &= 10
   -2x + 2y + 9z &= -13 \end{align} \(b\) takes the value for which the equations have an infinite number of solutions.
3. For the equations in part (ii) with the values of \(a\) and \(b\) found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations. [2]

At the beginning of the year John had a total of £2000 in three different accounts. He has twice as much money in the current account as in the savings account. • The current account has an interest rate of 2.5% per annum. • The savings account has an interest rate of 3.7% per annum. • The supersaver account has an interest rate of 4.9% per annum. John has predicted that he will earn a total interest of £92 by the end of the year.

1. Model this situation as a matrix equation. [2]
2. Find the amount that John had in each account at the beginning of the year. [2]
3. In fact, the interest John will receive is £92 **to the nearest pound**. Explain how this affects the calculations. [2]