4.03j Determinant 3x3: calculation

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

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]

The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} 1 & 4 & -1 \\ 3 & 0 & p \\ a & b & c \end{pmatrix},$$ where \(p\), \(a\), \(b\) and \(c\) are constants and \(a > 0\). Given that \(\mathbf{M}\mathbf{M}^T = k\mathbf{I}\) for some constant \(k\), find

1. the value of \(p\), [2]
2. the value of \(k\), [2]
3. the values of \(a\), \(b\) and \(c\), [6]
4. \(|\det \mathbf{M}|\). [2]

$$\mathbf{A} = \begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & k \end{pmatrix}.$$

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]

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]

By using the determinant of an appropriate matrix, find the values of \(\lambda\) for which the simultaneous equations \begin{align} 3x + 2y + 4z &= 5,
\lambda y + z &= 1,
x + \lambda y + \lambda z &= 4, \end{align} do not have a unique solution for \(x\), \(y\) and \(z\). [6]

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 \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & a & -1 \\ 0 & a & 2 \\ 1 & 2 & 1 \end{pmatrix}\).

1. Find, in terms of \(a\), the determinant of \(\mathbf{A}\). [3]
2. Three simultaneous equations are shown below. \begin{align} ax + ay - z &= -1
   ay + 2z &= 2a
   x + 2y + z &= 1 \end{align} For each of the following values of \(a\), determine whether the equations are consistent or inconsistent. If the equations are consistent, determine whether or not there is a unique solution.
   1. \(a = 0\)
   2. \(a = 1\)
   3. \(a = 2\) [6]

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A} = \begin{bmatrix} 2 & a & 3 \\ 0 & -2 & 1 \end{bmatrix} \quad \text{and} \quad \mathbf{B} = \begin{bmatrix} 1 & -3 \\ -2 & 4a \\ 0 & 5 \end{bmatrix}$$

1. Find the product \(\mathbf{AB}\) in terms of \(a\). [2 marks]
2. Find the determinant of \(\mathbf{AB}\) in terms of \(a\). [1 mark]
3. Show that \(\mathbf{AB}\) is singular when \(a = -1\) [2 marks]

The determinant \(A = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 0 & 2 \\ 3 & 2 & 1 \end{vmatrix}\) Which one of the determinants below has a value which is not equal to the value of \(A\)? Tick (\(\checkmark\)) one box. [1 mark] \(\begin{vmatrix} 3 & 1 & 3 \\ 2 & 0 & 2 \\ 3 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 1 & 2 & 3 \\ 1 & 0 & 2 \\ 1 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 2 & 2 & 2 \\ 1 & 0 & 1 \\ 3 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 0 & 2 \end{vmatrix}\) \quad \(\square\)

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

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]

Matrix M is given by \(\mathbf{M} = \begin{pmatrix} k & 1 & -5 \\ 2 & 3 & -3 \\ -1 & 2 & 2 \end{pmatrix}\), where \(k\) is a constant.

1. Show that \(\det \mathbf{M} = 12(k - 3)\). [2]
2. Find a solution of the following simultaneous equations for which \(x \neq z\). $$4x^2 + y^2 - 5z^2 = 6$$ $$2x^2 + 3y^2 - 3z^2 = 6$$ $$-x^2 + 2y^2 + 2z^2 = -6$$ [3]
3. 1. Verify that the point \((2, 0, 1)\) lies on each of the following three planes. $$3x + y - 5z = 1$$ $$2x + 3y - 3z = 1$$ $$-x + 2y + 2z = 0$$ [1]
   2. Describe how the three planes in part (iii) (A) are arranged in 3-D space. Give reasons for your answer. [4]
4. Find the values of \(k\) for which the transformation represented by M has a volume scale factor of 6. [3]

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 \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} \lambda & 1 & 14 \\ -1 & 2 & 8 \\ -3 & 2 & \lambda \end{pmatrix}\), where \(\lambda\) is a real constant.

1. Find an expression for the determinant of \(\mathbf{A}\) in terms of \(\lambda\). Give your answer in the form \(a\lambda^2 + b\lambda + c\), where \(a\), \(b\), \(c\) are integers whose values are to be determined. [3]
2. Show that \(\mathbf{A}\) is non-singular for all values of \(\lambda\). [4]

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

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]