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Quiz 3: Matrices

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Quiz

This quiz tests materials in the section called “Matrices”. If you are struggling with the questions go back and re-read the relevant material.

Define \(A= \begin{pmatrix}1&4 \\ 5&1 \\ 2&3\end{pmatrix}\)

Question 1: What is the order of \(A\)?

Choose the correct answer:

  1. \(3\)

  2. \(3 \times 2\)

  3. \(2 \times 3\)

  4. \(6\)

Question 2: What is the transpose of A?

Choose the correct answer:

  1. \(\begin{pmatrix} 1 &4 &5 \\2 &1 &3 \end{pmatrix}\)

  2. The transpose is undefined.

  3. \(\begin{pmatrix} 1 &5 &2 \end{pmatrix}\) and \(\begin{pmatrix} 4 &1 &3 \end{pmatrix}\)

  4. \(\begin{pmatrix} 1 &5 &2 \\ 4 &1 &3 \end{pmatrix}\)

Question 3: Re-write A as a partitioned matrix formed of two column vectors.

Choose the correct solution:

  1. \(A = \begin{pmatrix} a_1 &a_2 \end{pmatrix}\) where \(a_1 = \begin{pmatrix} 1 &5 &2 \end{pmatrix}'\) and \(a_2 = \begin{pmatrix} 4 &1 &3 \end{pmatrix}'\)

  2. \(A = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}\) where \(a_1 = \begin{pmatrix} 1 &5 &2 \end{pmatrix}\) and \(a_2 = \begin{pmatrix} 4 &1 &3 \end{pmatrix}\)

  3. Partitioning is only defined for square matrices.

  4. Partitioning requires the matrix to be split into four sub-matrices.

Define \(B= \begin{pmatrix} 4&2 &1 \\ 3&6 &3 \\ 2&5 &5\end{pmatrix}\)

Question 4: What is the main diagonal of B as a column vector?

Choose the correct answer:

  1. \(\begin{pmatrix} 4 &3 &2 \end{pmatrix}\)

  2. \(\begin{pmatrix} 4 &6 &5 \end{pmatrix}'\)

  3. \(\begin{pmatrix} 4 &2 &1 \end{pmatrix}'\)

  4. \(\begin{pmatrix} 4 &6 &5 \end{pmatrix}\)

Question 5: Calculate the trace of B.

Choose the correct answer:

  1. \(tr(B) = 16\)

  2. \(tr(B) = \begin{pmatrix} 4 &6 &5 \end{pmatrix}\)

  3. \(tr(B) = 15\)

  4. \(tr(B) = \begin{pmatrix} 4 &6 &5 \end{pmatrix}'\)

Define \(C= \begin{pmatrix} 2 &4 \\ 3 &1 \\ 6 &0\end{pmatrix}\)

Question 6: Calculate

(i) \(A+C\) (ii) \(A-C\) (iii) \((A+C)^T\) (iv) \(A^T + C^T\)

Choose the correct answer:

  1. \(A+C=\begin{pmatrix} 3&8\\8&2\\8&3\end{pmatrix}, \ \ A-C =\begin{pmatrix} -1&0\\2&0\\-4&3\end{pmatrix}, \ \ (A+C)^T=\begin{pmatrix} 3&8&8\\8&2&3\end{pmatrix}, \ \ A^T + C^T=\begin{pmatrix} 3&8\\8&2\\8&3\end{pmatrix}\)

  2. \(A+C=\begin{pmatrix} 3&8\\8&2\\8&3\end{pmatrix}, \ \ A-C =\begin{pmatrix} -1&0\\2&0\\4&3\end{pmatrix}, \ \ (A+C)^T=\begin{pmatrix} 3&8&8\\8&2&3\end{pmatrix}, \ \ A^T + C^T=\begin{pmatrix} 3&8&8\\8&2&3\end{pmatrix}\)

  3. \(A+C=\begin{pmatrix} 3&8\\8&2\\8&3\end{pmatrix}, \ \ A-C =\begin{pmatrix} -1&0\\2&0\\-4&3\end{pmatrix}, \ \ (A+C)^T=\begin{pmatrix} 3&8&8\\8&2&3\end{pmatrix}, \ \ A^T + C^T=\begin{pmatrix} 3&8&8\\8&2&3\end{pmatrix}\)

  4. \(A+C=\begin{pmatrix} 3&8\\8&2\\8&2\end{pmatrix}, \ \ A-C =\begin{pmatrix} -1&0\\2&0\\-4&3\end{pmatrix}, \ \ (A+C)^T=\begin{pmatrix} 3&8&8\\8&2&2\end{pmatrix}, \ \ A^T + C^T=\begin{pmatrix} 3&8\\8&2\\8&2\end{pmatrix}\)

Question 7: What is (a) \(2A\), and (b) \(-1((A-C)3)\)?

Choose the correct answer:

  1. \(2A = \begin{pmatrix} 2 &8 \\ 10&2 \\ 4&6\end{pmatrix}\) and \(-1((A-C)3) = \begin{pmatrix} 3&0\\-6&0\\12&-9\end{pmatrix}\)

  2. \(2A = \begin{pmatrix} 1&16\\25&1\\4&9\end{pmatrix}\) and \(-1((A-C)3) = \begin{pmatrix} 3&0\\-6&0\\12&-6\end{pmatrix}\)

  3. \(2A = \begin{pmatrix} 2 &8 \\ 10&2 \\ 4&6\end{pmatrix}\) and \(-1((A-C)3) = \begin{pmatrix}3&0\\6&0\\12&-9\end{pmatrix}\)

  4. \(2A = \begin{pmatrix} 1&16\\25&1\\4&9\end{pmatrix}\) and \(-1((A-C)3) = \begin{pmatrix} 3&0\\-6&1\\12&-9\end{pmatrix}\)

Define \(D= \begin{pmatrix} 3 \\ -1 \end{pmatrix}\). And we have defined \(A= \begin{pmatrix} 1&4\\5&1\\2&3 \end{pmatrix}\) above.

Question 8: What is the order of the matrix \(AD\)?

Choose the correct answer:

  1. \(3\)

  2. \(1 \times 3\)

  3. \(2 \times 3\)

  4. \(3 \times 1\)

Question 9: Calculate \(AD\).

Choose the correct answer:

  1. \(AD = \begin{pmatrix} -1 \\ 14 \\ 3 \end{pmatrix}^T\)

  2. \(AD = \begin{pmatrix} -1 \\ 14 \\ 3 \end{pmatrix}\)

  3. \(AD = \begin{pmatrix} -1 \\ 14 \\ -3 \end{pmatrix}\)

  4. None of the above

Remember that \(A= \begin{pmatrix}1&4\\5&1\\2&3\end{pmatrix}\) and \(B= \begin{pmatrix} 4&2 &1\\3&6 &3\\2&5 &5\end{pmatrix}\)

Question 10: What is the order of the matrix \(BA\)?

Choose the correct solution:

  1. \(3 \times 2\)

  2. \(3 \times 3\)

  3. \(2 \times 6\)

  4. \(2 \times 3\)

Question 11: Calculate \(BA\) and \(AB\).

Choose the correct solution:

  1. \(AB = \begin{pmatrix}16 &21 \\ 39&27 \\ 37 &28\end{pmatrix}\) and \(BA = \begin{pmatrix}16 &21 \\ 39&27 \\ 37 &28\end{pmatrix}\)

  2. \(AB\) is undefined. \(BA = \begin{pmatrix}16 &21 \\ 39&27 \\ 37 &28\end{pmatrix}\)

  3. \(BA\) is undefined. \(AB = \begin{pmatrix}16 &21 \\ 39&27 \\ 37 &28\end{pmatrix}\)

  4. \(BA\) and \(AB\) are both undefined.

Define \(E= \begin{pmatrix} 2 &1 \\ -3 &0 \end{pmatrix}\)

Question 12: Calculate the determinant of \(E\).

Choose the correct solution:

  1. -3

  2. It is undefined - \(E\) is a singular matrix.

  3. 3

  4. 9

Question 13: Calculate the inverse of \(E\).

Check your calculation by calculating \(E E^{-1}\).

Choose the correct solution:

  1. \(E^{-1} = \begin{pmatrix} 0 &-1/3 \\ 1&2/3\end{pmatrix}\)

  2. \(E^{-1} = \begin{pmatrix} 0 &1 \\ -1/3&2/3\end{pmatrix}\)

  3. \(E^{-1} = \begin{pmatrix} 0 &1/3 \\ -1&2/3\end{pmatrix}\)

  4. \(E^{-1} = \begin{pmatrix} 0 &-1 \\ 1/3&2/3\end{pmatrix}\)

Question 14: Is \(E\) (a) idempotent? and (b) orthogonal?

Choose the correct solution:

  1. \(E\) is idempotent but not orthogonal.

  2. \(E\) is both idempotent and orthogonal.

  3. \(E\) is not idempotent but it is orthogonal.

  4. \(E\) is neither idempotent nor orthogonal.

Quiz 2: Differentiation and integration Quiz 4: Combinatorics