Matrix Dimensions Exercise (with Solution)

Mention the dimension for each of the following matrix.

\[
\begin{array}{|c|c|}
\hline
\textbf{Question} & \textbf{Dimensions: ?} \\
\hline
A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} & ? \\
\hline
B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32} \end{pmatrix} & ? \\
\hline
C = \begin{pmatrix} c_{11} & c_{12} & c_{13} \\ c_{14} & c_{15} & c_{16} \end{pmatrix} & ? \\
\hline
D = \begin{pmatrix} d_{11} \\ d_{21} \\ d_{31} \end{pmatrix} & ? \\
\hline
E = \begin{pmatrix} e_{11} & e_{12} & e_{13} & e_{14} \\ e_{21} & e_{22} & e_{23} & e_{24} \\ e_{31} & e_{32} & e_{33} & e_{34} \end{pmatrix} & ? \\
\hline
F = \begin{pmatrix} f_{11} \end{pmatrix} & ? \\
\hline
\end{array}
\]

Solution:

\[
\begin{array}{|c|c|}
\hline
\textbf{Question} & \textbf{Explanation / Answer} \\
\hline
A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} & \text{Matrix \( A \) has dimensions \( 3 \times 3 \) (3 rows and 3 columns).} \\
\hline
B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32} \end{pmatrix} & \text{Matrix \( B \) has dimensions \( 3 \times 2 \) (3 rows and 2 columns).} \\
\hline
C = \begin{pmatrix} c_{11} & c_{12} & c_{13} \\ c_{14} & c_{15} & c_{16} \end{pmatrix} & \text{Matrix \( C \) has dimensions \( 2 \times 6 \) (2 rows and 6 columns).} \\
\hline
D = \begin{pmatrix} d_{11} \\ d_{21} \\ d_{31} \end{pmatrix} & \text{Matrix \( D \) has dimensions \( 3 \times 1 \) (3 rows and 1 column).} \\
\hline
E = \begin{pmatrix} e_{11} & e_{12} & e_{13} & e_{14} \\ e_{21} & e_{22} & e_{23} & e_{24} \\ e_{31} & e_{32} & e_{33} & e_{34} \end{pmatrix} & \text{Matrix \( E \) has dimensions \( 3 \times 4 \) (3 rows and 4 columns).} \\
\hline
F = \begin{pmatrix} f_{11} \end{pmatrix} & \text{Matrix \( F \) has dimensions \( 1 \times 1 \) (1 row and 1 column, i.e., a scalar).} \\
\hline
\end{array}
\]