4.03o Inverse 3x3 matrix

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]

\(\mathbf{M} = \begin{pmatrix} -1 & 2 & -1 \\ 2 & 2 & -2 \\ -1 & -2 & -1 \end{pmatrix}\)

1. Given that 4 is an eigenvalue of M, find a corresponding eigenvector. [3 marks]
2. Given that \(\mathbf{MU} = \mathbf{UD}\), where D is a diagonal matrix, find possible matrices for D and U. [8 marks]

The matrix A is \(\begin{pmatrix} -1 & 2 & 4 \\ 0 & -1 & -25 \\ -3 & 5 & -1 \end{pmatrix}\). Use the Cayley-Hamilton theorem to find A\(^{-1}\). [8]

The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 4 & 8 & 0 \\ 0 & \lambda & -2 \\ 4 & 0 & \lambda \end{pmatrix}.$$

1. Show that there are two values of \(\lambda\) for which \(\mathbf{A}\) is singular. [4]
2. Given that \(\lambda = 3\),
   1. determine the adjugate matrix of \(\mathbf{A}\),
   2. determine the inverse matrix \(\mathbf{A}^{-1}\). [5]

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]

$$\mathbf{A} = \begin{pmatrix} 2 & a \\ a-4 & b \end{pmatrix}$$ where \(a\) and \(b\) are non-zero constants. Given that the matrix \(\mathbf{A}\) is self-inverse,

1. determine the value of \(b\) and the possible values for \(a\). [5] The matrix \(\mathbf{A}\) represents a linear transformation \(M\). Using the smaller value of \(a\) from part (a),
2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line. [3]

Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = I + 2\mathbf{A}$$ where \(I\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]

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]

Given that the matrix \(\mathbf{A} = \begin{pmatrix} 2 & k \\ 1 & -3 \end{pmatrix}\), where \(k\) is real, is such that \(\mathbf{A}^3 = \mathbf{I}\), find the value of \(k\) and the numerical value of \(\det \mathbf{A}\). [4]

For real values of \(t\), the non-singular matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A}^{-1} = \begin{pmatrix} t & 5 \\ 2 & 8 \end{pmatrix} \quad \text{and} \quad \mathbf{B}^{-1} = \begin{pmatrix} 2 & -t \\ 3 & -1 \end{pmatrix}.$$

1. Determine the values which \(t\) cannot take. [2]
2. Without finding either \(\mathbf{A}\) or \(\mathbf{B}\), determine \((\mathbf{AB})^{-1}\) in terms of \(t\). [2]

0 & 2 & 2
- 1 & 1 & 3 \end{array} \right) .$$

1. Find the eigenvalues of \(\mathbf { A }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).