Matrix conformability and dimensions

1 Two small smooth spheres \(A\) and \(B\) have equal radii and have masses \(m\) and \(k m\) respectively. They are moving in a straight line in the same direction on a smooth horizontal table. The speed of \(A\) is \(u\) and the speed of \(B\) is \(\frac { 2 } { 3 } u\). Sphere \(A\) collides directly with sphere \(B\). The coefficient of restitution between the spheres is \(\frac { 4 } { 5 }\).

1. Show that the speed of \(A\) after the collision is \(\frac { u ( 2 k + 5 ) } { 5 ( k + 1 ) }\).
2. Given that the magnitude of the impulse experienced by \(B\) during the collision is \(\frac { 2 } { 5 } m u\), find the value of \(k\).

1

1. The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { l l l l } 1 & 0 & - 2 & 2 \\ 4 & 2 & - 2 & 3 \end{array} \right)\).
   1. Write down the dimensions of \(\mathbf { P }\).
   2. Write down the transpose of \(\mathbf { P }\).
2. The matrices \(\mathbf { Q } , \mathbf { R }\) and \(\mathbf { S }\) are given by \(\mathbf { Q } = \left( \begin{array} { l l } 1 & 2 \end{array} \right) , \mathbf { R } = \left( \begin{array} { r r } 3 & - 4 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { S } = \left( \begin{array} { l l } 3 & - 2 \end{array} \right)\). Write down the sum of the two of these matrices which are conformable for addition.
3. The dimensions of matrix \(\mathbf { A }\) are 4 by 5. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are conformable for multiplication so that the matrix \(\mathbf { C } = \mathbf { B A }\) can be formed. The matrix \(\mathbf { C }\) has 6 rows.
   1. Write down the number of columns that \(\mathbf { C }\) has.
   2. Write down the dimensions of \(\mathbf { B }\).
   3. Explain whether the matrix \(\mathbf { A B }\) can be formed.
4. Find the value of \(c\) for which \(\left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right) \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) = \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) \left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right)\).

2 Matrices \(\mathbf { P }\) and \(\mathbf { Q }\) are given by \(\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0 \\ - 2 & 1 & 3 \end{array} \right)\) and \(\mathbf { Q } = ( ( 1 + k ) - 1 )\) where \(k\) is a constant.
Exactly one of statements A and B is true.
Statement A: \(\quad \mathbf { P }\) and \(\mathbf { Q }\) (in that order) are conformable for multiplication.
Statement B: \(\quad \mathbf { Q }\) and \(\mathbf { P }\) (in that order) are conformable for multiplication.

1. State, with a reason, which one of A and B is true.
2. Find either \(\mathbf { P Q }\) or \(\mathbf { Q P }\) in terms of \(k\).

1 Matrices \(\mathbf { P }\) and \(\mathbf { Q }\) are given by \(\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0 \\ - 2 & 1 & 3 \end{array} \right)\) and \(\mathbf { Q } = ( ( 1 + k ) - 1 )\) where \(k\) is a constant.
Exactly one of statements A and B is true.
Statement A: \(\quad \mathbf { P }\) and \(\mathbf { Q }\) (in that order) are conformable for multiplication.
Statement B: \(\quad \mathbf { Q }\) and \(\mathbf { P }\) (in that order) are conformable for multiplication.

1. State, with a reason, which one of A and B is true.
2. Find either \(\mathbf { P Q }\) or \(\mathbf { Q P }\) in terms of \(k\).

Three matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are given by $$\mathbf{A} = \begin{pmatrix} 5 & 2 & -3 \\ 0 & 7 & 6 \\ 4 & 1 & 0 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 1 & 0 \\ 3 & -5 \\ -2 & 6 \end{pmatrix} \quad \text{and } \mathbf{C} = \begin{pmatrix} 6 & 4 & 3 \\ 1 & 2 & 0 \end{pmatrix}$$ Which of the following **cannot** be calculated? Circle your answer. [1 mark] \(\mathbf{AB}\) \(\qquad\) \(\mathbf{AC}\) \(\qquad\) \(\mathbf{BC}\) \(\qquad\) \(\mathbf{A}^2\)

Which of the following matrices is an identity matrix? Circle your answer. [1 mark] \(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\)