Matrix conformability and dimensions

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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 \(\left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right) \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) = \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) \left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right)\).

1 Which of the following matrices is an identity matrix?
Circle your answer. $$\left[ \begin{array} { l l } 1 & 1 \\ 1 & 1 \end{array} \right] \quad \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 0 \\ 0 & 0 \end{array} \right]$$

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 \(k\).