Theory:

In this section, let us discuss different types of matrices.
1. Row matrix
A matrix is a row matrix when it is made up of just one row and '\(n\)' number of columns. Row matrices are also called row vectors.
For a matrix \(A = \begin{bmatrix}
a_{11} & a_{12} & a_{13} & … & a_{1n}
\end{bmatrix}\), the order of the matrix is \(1\) \(\times\) \(n\).
Example:
Let us look at a few row matrices.
 
1. \(A = \begin{bmatrix}
\sqrt{2} & \frac{\sqrt 7}{2} & 12
\end{bmatrix}\), where the order of the matrix is \(1\) \(\times\) \(3\).
 
2. \(A = \begin{bmatrix}
7 & 14 & 21 & 28 & 35 & 42 & 49
\end{bmatrix}\), where the order of the matrix is \(1\) \(\times\) \(7\).
2. Column matrix
A matrix is a column matrix when it is made up of 'm' number of rows and just one column. Column matrices are also called column vectors.
For a matrix \(A = \begin{bmatrix}
a_{11}\\
a_{21}\\
a_{31}\\
\vdots\\
a_{m1}
\end{bmatrix}\), the order of the matrix is \(m\) \(\times\) \(1\).
Example:
Let us look at a few column matrices.
 
1. \(A = \begin{bmatrix}
2x\\
3x\\
x
\end{bmatrix}\), where the order of the matrix is \(3\) \(\times\) \(1\).
 
2. \(A = \begin{bmatrix}
4\\
8\\
12\\
16\\
20
\end{bmatrix}\), where the order of the matrix is \(5\) \(\times\) \(1\).
3. Square matrix
A matrix is a square matrix when the number of rows equals the number of columns. In other words, \(m\) \(=\) \(n\).
The order of a square matrix is \(m\).
Example:
Let us look at a few square matrices.
 
1. \(A = \begin{bmatrix}
1 & 2\\
3 & 4
\end{bmatrix}\), where the order of the matrix is \(2\) \(\times\) \(2\).
 
2. \(A = \begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9
\end{bmatrix}\), where the order of the matrix is \(3\) \(\times\) \(3\).
Leading diagonal of a square matrix:
In a square matrix, the entries \(a_{ij}\) where \(i\) \(=\) \(j\) form the leading diagonal of a matrix.
Example:
Let us consider the matrix given below.
 
\(A = \begin{bmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{bmatrix}\)
 
Here, the elements \(a_{11}\), \(a_{22}\) and \(a_{33}\) form the elements of the leading diagonal.
4. Diagonal matrix
In a square matrix, when all the entries except the leading diagonal is zero, then it is called a diagonal matrix. In other words, \(a_{ij}\) \(=\) \(0\) for \(i\) \(\neq\) \(j\).
Example:
Let us look at a few diagonal matrices.
 
1. \(A = \begin{bmatrix}
1 & 0\\ 
0 & 4
\end{bmatrix}\)
 
2. \(A = \begin{bmatrix}
1 & 0 & 0\\ 
0 & 5 & 0\\ 
0 & 0 & 9
\end{bmatrix}\)
5. Scalar matrix
In a square matrix, when all the elements of the leading diagonal is the same, then it is a scalar matrix
The general representation of a scalar matrix \(A\) \(=\) \((a_{ij})_{m \times n}\) is \(a_{ij} = \begin{cases}
0 & \text{ when } i \neq j \\
k & \text{ when } i = j
\end{cases}\), where \(k\) is a constant.
Example:
Let us look at a few scalar matrices.
 
1. \(A = \begin{bmatrix}
7 & 0 & 0\\ 
0 & 7 & 0\\ 
0 & 0 & 7
\end{bmatrix}\)
 
2. \(A = \begin{bmatrix}
\sqrt{3} & 0 & 0\\ 
0 & \sqrt{3} & 0\\ 
0 & 0 & \sqrt{3}
\end{bmatrix}\)
6. Identity or unit matrix
In a square matrix, when all the leading diagonal elements are \(1\), then it is an identity matrix or a unit matrix.
The general representation of a unit matrix \(A\) \(=\) \((a_{ij})_{m \times n}\) is \(a_{ij} = \begin{cases}
0 & \text{ when } i \neq j \\ 
1 & \text{ when } i = j
\end{cases}\).
Example:
Let us look at a few identity matrices.
 
1. \(\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}\)
 
2. \(\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}\)
7. Zero matrix or null matrix
A matrix is a zero matrix or a null matrix when all the matrix elements are zero.
Example:
Let us look at a few null matrices.
 
1. \(\begin{bmatrix}
0 & 0\\
0 & 0
\end{bmatrix}\)
 
2. \(\begin{bmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}\)
8. Transpose of a matrix
The transpose of a matrix is obtained by interchanging the elements in the rows and columns. The transpose of a matrix is denoted by \(A^T\), and \(A^T\) is read as '\(A\) \(\text{transpose}\)'.
If the order of matrix \(A\) is \( m \times n\), then the order of \(A^T\) is \(n \times m\).
Example:
Let us now look at a few examples.
 
1. If \(A = \begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9
\end{bmatrix}\), then \(A^T = \begin{bmatrix}
1 & 4 & 7\\
2 & 5 & 8\\
3 & 6 & 9
\end{bmatrix}\).
 
2. If \(B = \begin{bmatrix}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12
\end{bmatrix}\), then \(B^T = \begin{bmatrix}
1 & 5 & 9\\
2 & 6 & 10\\
3 & 7 & 11\\
4 & 8 & 12
\end{bmatrix}\).
9. Triangular matrix
If all the entries above the leading diagonal are zero in a square matrix, then it is a lower triangular matrix. If all the entries below the leading diagonal are zero, then it is an upper triangular matrix.
A matrix \(A\) is a upper triangular matrix if \(a_{ij}\) \(=\) \(0\) for \(i\) \(>\) \(j\). Similarly, matrix \(A\) is a lower triangular matrix if \(a_{ij}\) \(=\) \(0\) for \(i\) \(<\) \(j\).
Example:
Let us look at a few examples.
 
Upper triangular matrices:
 
\(A = \begin{bmatrix}
1 & 0 & 0\\
2 & 3 & 0\\
4 & 5 & 6
\end{bmatrix}\),   \(B = \begin{bmatrix}
7 & 0 & 0\\
8 & 9 & 0\\
10 & 11 & 12
\end{bmatrix}\)
 
Lower triangular matrices:
 
\(C = \begin{bmatrix}
1 & 2 & 3\\ 
0 & 4 & 5\\ 
0 & 0 & 6
\end{bmatrix}\),   \(D = \begin{bmatrix}
7 & 8 & 9\\ 
0 & 10 & 11\\ 
0 & 0 & 12
\end{bmatrix}\)