Solving linear systems using matrices

7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c } a & 3 \\ - 2 & 1 \end{array} \right)\).

1. Given that \(\mathbf { A }\) is singular, find \(a\).
2. Given instead that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the simultaneous equations $$\begin{aligned} a x + 3 y & = 1 \\ - 2 x + y & = - 1 \end{aligned}$$

1

1. Find the inverse of the matrix \(\mathbf { A } = \left( \begin{array} { l l } 4 & 3 \\ 1 & 2 \end{array} \right)\).
2. Use this inverse to solve the simultaneous equations $$\begin{aligned} 4 x + 3 y & = 5 \\ x + 2 y & = - 4 \end{aligned}$$ showing your working clearly.

1

1. Find the inverse of the matrix \(\mathbf { M } = \left( \begin{array} { r r } 4 & - 1 \\ 3 & 2 \end{array} \right)\).
2. Use this inverse to solve the simultaneous equations $$\begin{aligned} & 4 x - y = 49 \\ & 3 x + 2 y = 100 \end{aligned}$$ showing your working clearly.

2 You are given that \(\mathbf { M } = \left( \begin{array} { r r } 2 & - 5 \\ 3 & 7 \end{array} \right)\). \(\mathbf { M } \binom { x } { y } = \binom { 9 } { - 1 }\) represents two simultaneous equations.

1. Write down these two equations.
2. Find \(\mathbf { M } ^ { - 1 }\) and use it to solve the equations.

9 The simultaneous equations $$\begin{aligned} & 2 x - y = 1 \\ & 3 x + k y = b \end{aligned}$$ are represented by the matrix equation \(\mathbf { M } \binom { x } { y } = \binom { 1 } { b }\).

1. Write down the matrix \(\mathbf { M }\).
2. State the value of \(k\) for which \(\mathbf { M } ^ { - 1 }\) does not exist and find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) when \(\mathbf { M } ^ { - 1 }\) exists. Use \(\mathbf { M } ^ { - 1 }\) to solve the simultaneous equations when \(k = 5\) and \(b = 21\).
3. What can you say about the solutions of the equations when \(k = - \frac { 3 } { 2 }\) ?
4. The two equations can be interpreted as representing two lines in the \(x - y\) plane. Describe the relationship between these two lines
   (A) when \(k = 5\) and \(b = 21\),
   (B) when \(k = - \frac { 3 } { 2 }\) and \(b = 1\),
   (C) when \(k = - \frac { 3 } { 2 }\) and \(b = \frac { 3 } { 2 }\). RECOGNISING ACHIEVEMENT

1 Given that \(\mathbf { M } \binom { x } { y } = \binom { 1 } { 3 }\), where \(\mathbf { M } = \left( \begin{array} { r r } 4 & - 3 \\ 8 & 21 \end{array} \right)\), find \(x\) and \(y\).

4 You are given the system of equations $$\begin{array} { r } a ^ { 2 } x - 2 y = 1 \\ x + b ^ { 2 } y = 3 \end{array}$$ where \(a\) and \(b\) are real numbers.

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

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 2 & - 2 \\ 1 & 3 \end{array} \right)\).

1. Calculate \(\operatorname { det } \mathbf { A }\).
2. Write down \(\mathbf { A } ^ { - 1 }\).
3. Hence solve the equation \(\mathbf { A } \binom { \mathrm { x } } { \mathrm { y } } = \binom { - 1 } { 2 }\).
4. Write down the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = 4 \mathbf { I }\). Matrices \(\mathbf { C }\) and \(\mathbf { D }\) are given by \(\mathbf { C } = \left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { l l l } 0 & 2 & p \end{array} \right)\) where \(p\) is a constant.
5. Find, in terms of \(p\),
   • the matrix CD
   • the matrix DC.
   It is observed that \(\mathbf { C D } \neq \mathbf { D C }\).
6. The result that \(\mathbf { C D } \neq \mathbf { D C }\) is a counter example to the claim that matrix multiplication has a particular property. Name this property.

1

1. 1. Write the following simultaneous equations as a matrix equation. $$\begin{aligned} x + y + 2 z & = 7 \\ 2 x - 4 y - 3 z & = - 5 \\ - 5 x + 3 y + 5 z & = 13 \end{aligned}$$
   2. Hence solve the equations.
2. Determine the set of values of the constant \(k\) for which the matrix equation $$\left( \begin{array} { c c } k + 1 & 1 \\ 2 & k \end{array} \right) \binom { x } { y } = \binom { 23 } { - 17 }$$ has a unique solution.

4. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 4 \\ k & 2 & - 2 \\ 4 & 1 & - 2 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r r } k - 7 & 6 & - 10 \\ 2 & - 20 & 24 \\ - 3 & 2 & - 1 \end{array} \right)$$ where \(k\) is a constant.

1. Determine, in simplest form in terms of \(k\), the matrix \(\mathbf { M N }\).
2. Given that \(k = 5\)
   1. write down \(\mathbf { M N }\)
   2. hence write down \(\mathbf { M } ^ { - 1 }\)
3. Solve the simultaneous equations $$\begin{aligned} & 2 x + y + 4 z = 2 \\ & 5 x + 2 y - 2 z = 3 \\ & 4 x + y - 2 z = - 1 \end{aligned}$$
4. Interpret the answer to part (c) geometrically.

4. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & - 2 & - 7 \\ 3 & k & 2 \\ 1 & 1 & 4 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { c c c } 4 k - 2 & 1 & 7 k - 4 \\ - 10 & 3 & - 19 \\ 3 - k & - 1 & 6 - k \end{array} \right)$$ where \(k\) is a constant.

1. Determine the value of the constant \(c\) for which $$\mathbf { A B } = ( 3 k + c ) \mathbf { I }$$
2. Hence determine the value of \(k\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Given that \(\mathbf { A } ^ { - 1 }\) does exist,
3. write down \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
4. Use the answer to part (c) to solve the simultaneous equations $$\begin{aligned} - x - 2 y - 7 z & = 10 \\ 3 x + k y + 2 z & = 3 \\ x + y + 4 z & = 1 \end{aligned}$$ giving the values of \(x , y\) and \(z\) in simplest form in terms of \(k\).

1. Tyler invested a total of \(\pounds 5000\) across three different accounts; a savings account, a property bond account and a share dealing account.

Tyler invested \(\pounds 400\) more in the property bond account than in the savings account.
After one year

• the savings account had increased in value by \(1.5 \%\)
• the property bond account had increased in value by \(3.5 \%\)
• the share dealing account had decreased in value by \(2.5 \%\)
• the total value across Tyler's three accounts had increased by \(\pounds 79\)

Form and solve a matrix equation to find out how much money was invested by Tyler in each account.

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

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