Row Matrix

Row Matrix: A rectangular array of numbers that are arranged in rows and columns is known as a "matrix." The size of a matrix can be determined by the number of rows and columns in it. If a matrix has "m" rows and "n" columns, then it is said to be an "m by n" matrix and is written as an "m × n" matrix.

In mathematics, a row matrix is a matrix characterized by having only one row, though it may contain multiple columns. If the matrix is in the form of 1 x n, where 'n' represents the number of columns, it qualifies as a row matrix. Its elements are arranged in such a way that they form a single horizontal row within the matrix structure.

For example, if a matrix has five rows and three columns, it is a "5 × 3" matrix. We have various types of matrices, like rectangular, square, triangular, symmetric, singular, etc. Now let us discuss the types of matrices in detail.

What is a Row Matrix?

A row matrix is defined as a matrix that has only one row. A matrix "A = [a_ij]" is said to be a row matrix if the order of the matrix is "1 × n." In a row matrix, all the entries are arranged in a single row. A row matrix can have numerous columns but only one row. For example, the matrix given below is a row matrix of order "1 × 3," which has one row and three columns that are equal to the number of entries in the matrix.

Row Matrix Definition

A row matrix, also known as a row vector, is a matrix with a single row. It's a way to represent data in a horizontal format, where each element of the row represents a different variable or value.

Row Matrix Examples

• The matrix given below is a row matrix of order "1 × 3."

A_{1\times3} = \left[\begin{array}{cccc} p & q & r\end{array}\right]

• The matrix given below is a row matrix of order "1 × 4."

B_{1\times4} = \left[\begin{array}{cccc} a & b & c & d\end{array}\right]

• The matrix given below is a row matrix of order "1 × 5."

C_{1\times5} = \left[\begin{array}{ccccc} 0 & 5 & 8 & -7 & 11\end{array}\right]

Properties of a Row Matrix

The following are some important properties of a row matrix:

• Any row matrix will have only one row.
• A row matrix can have numerous columns.
• The number of entries in a row matrix is equal to the number of columns.
• A row matrix is also a rectangular matrix and a horizontal matrix.
• The transpose of a row matrix is a column matrix and vice versa.
• Any two-row matrices can be added or subtracted if the order of both matrices is the same.
• The multiplication of a row matrix is possible only with a column matrix if and only if the number of rows in the column matrix is equal to the number of columns in the given row matrix.
• A singleton matrix is obtained when a row matrix and column matrix are multiplied.

Operations on a Row Matrix

We can operate on row matrix using operations such as:

• Addition
• Subtraction

Addition of Row Matrices

Any two-row matrices can be added if the order of both matrices is the same. If the orders of both matrices are the same, then the corresponding entries are added.

For example, let M = \left[\begin{array}{ccc} -2 & 7 & 11\end{array}\right]_{1\times3} and N = \left[\begin{array}{ccc} 5 & 0 & -6\end{array}\right]_{1\times3}

M + N = \left[\begin{array}{ccc} -2 & 7 & 11\end{array}\right]_{1\times3} + \left[\begin{array}{ccc} 5 & 0 & -6\end{array}\right]_{1\times3}

M + N = \left[\begin{array}{ccc} (-2+5) & (7+0) & (11+(-6))\end{array}\right]_{1\times3}

M + N = \left[\begin{array}{ccc} 3 & 7 & 5\end{array}\right]_{1\times3}

Subtraction of Row Matrices

Any two-row matrices can be subtracted if the order of both matrices is the same. If the orders of both matrices are the same, then the corresponding entries are subtracted.

For example, let A = \left[\begin{array}{ccc} 22 & -13 & 19\end{array}\right]_{1\times3} and B = \left[\begin{array}{ccc} 12 & -8 & 7\end{array}\right]_{1\times3}

A - B = \left[\begin{array}{ccc} 22 & -13 & 19\end{array}\right]_{1\times3} + \left[\begin{array}{ccc} 12 & -8 & 7\end{array}\right]_{1\times3}

A - B = \left[\begin{array}{ccc} 10 & -5 & 12\end{array}\right]_{1\times3}

Multiplication

The multiplication of a row matrix is possible only with a column matrix if and only if the number of rows in the column matrix is equal to the number of columns in the given row matrix.

For example, let P = \left[\begin{array}{ccc} 1 & 2 & 3\end{array}\right]_{1\times3}and Q = \left[\begin{array}{c} 4\\ 0\\ 5 \end{array}\right]_{3\times1}

P × Q = \left[\begin{array}{ccc} 1 & 2 & 3\end{array}\right]_{1\times3}\times\left[\begin{array}{c} 4\\ 0\\ 5 \end{array}\right]_{3\times1}

P × Q = [19]_1×1

Row Matrix Multiplied by Column Matrix

When you multiply a row matrix by a column matrix, you perform what's called a dot product or inner product operation.

Row and Column Matrix

The common difference between row and column matrix is listed in the following table:

People Also Read:

• Matrices
• Types of Matrices
• Order of Matrix
• Matrix Operations

Solved Examples on Row Matrices

Example 1: Find the transpose of the matrix given below.

Solution:

The matrix given matrix is a row matrix of order "1 × 4." We know that the transpose of a matrix is obtained by interchanging the entries of rows and columns. So, the order of the transpose of the given matrix will be "4 × 1."

A = \left[\begin{array}{cccc} a & b & c & d\end{array}\right]_{1\times4} ⇒ A^{T} = \left[\begin{array}{c} a\\ b\\ c\\ d \end{array}\right]_{4\times1}

Example 2: Find the value of A + 2B, if A = \left[\begin{array}{cccc} 3 & 7 & 0 & -2\end{array}\right]_{1\times4} and B = \left[\begin{array}{cccc} 1 & -3 & 5 & 1\end{array}\right]_{1\times4}

Solution:

A + 2B = \left[\begin{array}{cccc} 3 & 7 & 0 & -2\end{array}\right]_{1\times4}+ 2 \times\left[\begin{array}{cccc} 1 & -3 & 5 & 1\end{array}\right]_{1\times4}

A + 2B = \left[\begin{array}{cccc} 3 & 7 & 0 & -2\end{array}\right]_{1\times4}+ \left[\begin{array}{cccc} 2 & -6 & 10 & 2\end{array}\right]_{1\times4}

A + 2B = \left[\begin{array}{cccc} (3+2) & (7-6) & (0+10) & (-2+2)\end{array}\right]_{1\times4}

A + 2B = \left[\begin{array}{cccc} 5 & 1 & 10 & 0\end{array}\right]_{1\times4}

Example 3: Find the product of the matrices given below.

M = \left[\begin{array}{ccc} 11 & -5 & 8\end{array}\right]_{1\times3}and N = \left[\begin{array}{c} 2\\ 3\\ 4 \end{array}\right]_{3\times1}

Solution:

M × N = \left[\begin{array}{ccc} 11 & -5 & 8\end{array}\right]_{1\times3}\times\left[\begin{array}{c} 2\\ 3\\ 4 \end{array}\right]_{3\times1}

M × N = \left[\begin{array}{c} (11\times2)+(-5\times3)+(8\times4)\end{array}\right]_{1\times1}

M × N = \left[\begin{array}{c} 39\end{array}\right]_{1\times1}

Example 4: Find the value of P − Q, if P = \left[\begin{array}{cc} 25 & 14\end{array}\right]_{1\times2}and Q = \left[\begin{array}{cc} 13 & 10\end{array}\right]_{1\times2}

Solution:

P - Q = \left[\begin{array}{cc} 25 & 14\end{array}\right]_{1\times2}- \left[\begin{array}{cc} 13 & 10\end{array}\right]_{1\times2}

P - Q = \left[\begin{array}{cc} (25-13) & (14-10)\end{array}\right]_{1\times2}

P - Q = \left[\begin{array}{cc} 12 & 4\end{array}\right]_{1\times2}