Types of Matrices Based on Elements
1. Null Matrix
- A matrix where all elements are zero is called a null matrix.
- Example:
\[\Large
Null\;Matrix\;C =
\begin{bmatrix}
\color{blue}{0} & \color{blue}{0} & \color{blue}{0} \\
\color{blue}{0} & \color{blue}{0} & \color{blue}{0} \\
\color{blue}{0} & \color{blue}{0} & \color{blue}{0}
\end{bmatrix}
\]
- Null matrices are denoted by 0.
2. Diagonal Matrix
- A square matrix where all elements except those on the leading diagonal are zero.
- Example:
\[\Large
Diagonal\;Matrix\;D =
\begin{bmatrix}
\color{red}{2} & \color{blue}{0} & \color{blue}{0} \\
\color{blue}{0} & \color{red}{5} & \color{blue}{0} \\
\color{blue}{0} & \color{blue}{0} & \color{red}{8}
\end{bmatrix}
\]
- Only the elements 2, 5, 8 are on the diagonal.
3. Unit Matrix
- A diagonal matrix where all diagonal elements are 1.
- Also called the identity matrix.
- Example:
\[\Large
Unit\;Matrix\;I =
\begin{bmatrix}
\color{red}{1} & \color{blue}{0} & \color{blue}{0} \\
\color{blue}{0} & \color{red}{1} & \color{blue}{0} \\
\color{blue}{0} & \color{blue}{0} & \color{red}{1}
\end{bmatrix}
\]
- It behaves like the number 1 in arithmetic calculations.
\[\Large A \cdot I = A\]
Let \( A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \) and \( I = \begin{bmatrix} \color{red}{1} & \color{blue}{0} \\ \color{blue}{0} & \color{red}{1} \end{bmatrix} \).
Multiplying \( A \) by \( I \) gives:
\[\Large A \cdot I = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \cdot \begin{bmatrix} \color{red}{1} & \color{blue}{0} \\ \color{blue}{0} & \color{red}{1} \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \]- As expected multiplying a Matrix by a Unit Matrix will give us the same Matrix
Which of the following is a characteristic of a null matrix?