How to Solve Questions on Row Matrices for Exams
A row Matrix is just a mathematical way of representing a list of numbers in horizontal form. We may encounter different sets of data which can be represented in a list or an array form, such as the price of different stationary items or marks got by a student in different subjects and many more. Writing this data in the form of a Row Matrix makes it simple to read and understand the data.
What is Row Matrix?
A Row Matrix is a matrix having a single row. Recall that a Matrix is a rectangular array of numbers or symbols which is used to represent some physical object, having numbers arranged in rows and columns. In mathematical form, A Matrix having m rows and n columns is represented by
\[A = {\left[ {{a_{ij}}} \right]_{m \times n}}\]
So, a Row Matrix will have only one row and n columns. Hence, a Row Matrix is represented by \[{[{a_{1j}}]_{1 \times n}}\]where \[{a_{1j}}\]is the (1,j)th entry of the matrix.
Representation and Size
A Row Matrix is written in the form:
\[{[{a_{1j}}]_{1 \times n}}\]where \[{a_{1j}}\]is the (1,j)th entry of the matrix. The number of entries of the matrix \[\left[a_{1j}\right]_{1 \times n}\] is n. The number of entities in a row matrix depends on the number of columns. In other words, the number of entities of a row matrix is equal to the number of columns. A Row Matrix always has only one row. It can have any number of columns. We say that a row Matrix has n columns. So, a Row Matrix is n-dimensional.
Examples of Row Matrix
A Row Matrix of order 1×2 is \[\left[ {\begin{array}{*{20}{c}}2&5\end{array}} \right]\].
A Row Matrix of size 5 is \[\left[ {\begin{array}{*{20}{c}}{2.5}&{3.6}&{5.1}&{ - 2.0}&{6.3}\end{array}} \right]\].
Row Matrix containing 3 elements is \[\left[ {\begin{array}{*{20}{c}}{60}&{40}&{50}\end{array}} \right]\].
Row Matrix of order 1×4 with alternative entries 1 and 0 is \[\left[ {\begin{array}{*{20}{c}}1&0&1&0\end{array}} \right]\].
Practical Uses of Row Matrix
While sending a message over a network, the sender device encodes it into a binary format using an array of 0's and 1's in Row Matrix Form. This is known as Cryptography.
We can store data such as the Family's total monthly cost using a one-dimensional horizontal matrix, i.e., a Row Matrix.
Properties of Row Matrix
Commutativity
\[A + B = B + A\]
Associativity
\[A + \left( {B + C} \right) = \left( {A + B} \right) + C\]
Operations on Row Matrix
1. Addition of Row Matrices
We can add two row matrices by simply adding their corresponding entries.
This is depicted as:
Given two matrices \[A = \left[ {\begin{array}{*{20}{c}}1&2&3\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{c}}2&3&4\end{array}} \right]\], We need to find \[C = A + B\]. Implies
\[\begin{array}{l}C = \left[ {\begin{array}{*{20}{c}}1&2&3\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}2&3&4\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{1 + 2}&{2 + 3}&{3 + 4}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}3&5&7\end{array}} \right]\end{array}\]
So, the addition of two matrices is \[\left[ {\begin{array}{*{20}{c}}3&5&7\end{array}} \right]\].
Similarly, we can add three matrices together by adding their corresponding entries as:
Given \[P = \left[ {\begin{array}{*{20}{c}}2&{2.5}&3&{1.5}\end{array}} \right]\], \[Q = \left[ {\begin{array}{*{20}{c}}{1.5}&{ - 1}&{ - 4.25}&5\end{array}} \right]\], \[R = \left[ {\begin{array}{*{20}{c}}0&{ - 1.5}&1&{ - 2.5}\end{array}} \right]\]. We need to find \[S = P + Q + R\]. Putting values of P, Q, and R, we get
\[S = \left[ {\begin{array}{*{20}{c}}2&{2.5}&3&{1.5}\end{array}\left] + \right[\begin{array}{*{20}{c}}{1.5\;\,}&{ - 1}&{ - 4.25}&5\end{array}\left] + \right[\begin{array}{*{20}{c}}{0\;}&{ - 1.5}&1&{ - 2.5}\end{array}} \right]\]
\[ = \left[ {\begin{array}{*{20}{c}}{2 + 1.5 + 0\;}&{2.5 + \left( { - 1} \right) + \left( { - 1.5} \right)}&{3 + \left( { - 4.25} \right) + 1\;}&{1.5 + 5 + \left( { - 2.5} \right)}\end{array}\;} \right]\]
\[ = {\rm{ }}\left[ {\begin{array}{*{20}{c}}{3.5}&0&{ - 0.25}&{4.0}\end{array}} \right]\]
2. Subtraction of Row Matrices
We can subtract one matrix from another by subtracting their corresponding ith entries. This is depicted as:
Given two matrices \[A = \left[ {\begin{array}{*{20}{c}}1&2&3\end{array}} \right]\] and\[B = \left[ {\begin{array}{*{20}{c}}2&3&4\end{array}} \right]\]. We need to find \[D = A - B\]. Implies
\[\begin{array}{l}D = \left[ {\begin{array}{*{20}{c}}1&2&3\end{array}} \right]{\rm{ }} - \left[ {\begin{array}{*{20}{c}}2&3&4\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{1 - 2}&{2 - 3}&{3 - 4}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 1}&{ - 1}\end{array}} \right]\end{array}\]
So, the subtraction of second matrix from first is \[\left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 1}&{ - 1}\end{array}} \right]\].
Similarly, we can subtract two matrices from a given matrix together as:
Given \[P = \left[ {\begin{array}{*{20}{c}}2&{2.5}&3&{1.5}\end{array}} \right]\], \[Q = \left[ {\begin{array}{*{20}{c}}{1.5}&{ - 1}&{ - 4.25}&5\end{array}} \right]\], \[R = \left[ {\begin{array}{*{20}{c}}0&{ - 1.5}&1&{ - 2.5}\end{array}} \right]\]. We need \[T = P - Q - R\]. Putting values of P, Q, and R, we get
\[\begin{array}{l}S = \left[ {\begin{array}{*{20}{c}}2&{2.5}&3&{1.5}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}{1.5}&{ - 1}&{ - 4.25}&5\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}0&{ - 1.5}&1&{ - 2.5}\end{array}} \right]{\rm{ }}\\ = \left[ {\begin{array}{*{20}{c}}{2 - 1.5 - 0\;}&{2.5 - \left( { - 1} \right) - \left( { - 1.5} \right)}&{3 - \left( { - 4.25} \right) - 1\;}&{1.5 - 5 - \left( { - 2.5} \right)}\end{array}\;} \right]\\ = \left[ {\begin{array}{*{20}{c}}{0.5}&{5.0}&{6.25}&{ - 1}\end{array}{\rm{.0}}} \right]\end{array}\].
This can also be done as \[T = P - \left( {Q + R} \right)\].
Put \[Q + R = M\], we get
\[\begin{array}{*{20}{l}}{M = \left[ {\begin{array}{*{20}{c}}{1.5 + 0}&{ - 1 + \left( { - 1.5} \right)}&{\left( { - 4.25} \right) + 1}&{{\rm{ }}5 + \left( { - 2.5} \right)}\end{array}} \right]}\\{ = \left[ {\begin{array}{*{20}{c}}{1.5}&{ - 2.5}&{ - 3.25}&{2.5}\end{array}} \right]}\end{array}\]
Now,
\[\begin{array}{*{20}{l}}{S = P - M}\\\begin{array}{l} = \left[ {\begin{array}{*{20}{c}}2&{2.5}&3&{1.5}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}{1.5}&{ - 2.5}&{ - 3.25}&{2.5}\end{array}} \right]{\rm{ }}\\ = \left[ {\begin{array}{*{20}{c}}{0.5}&{5.0}&{6.25}&{ - 1.0}\end{array}} \right]\end{array}\end{array}\]
3. Multiplication of Row Matrix by a Scalar
We can multiply a matrix by a scalar as: Given \[B = [\begin{array}{*{20}{c}}{{b_{11}}}&{{b_{12}}}&{{b_{13}}}& \ldots &{{b_{1n}}}\end{array}]\]. Multiplying B by a scalar k, we get \[kB = k[\begin{array}{*{20}{c}}{{b_{11}}}&{{b_{12}}}&{{b_{13}}}& \ldots &{{b_{1n}}}\end{array}]\]
\[kB = [\begin{array}{*{20}{c}}{k{b_{11}}}&{k{b_{12}}}&{k{b_{13}}}& \ldots &{k{b_{1n}}}\end{array}]\].
For example:
\[\begin{array}{*{20}{l}}\begin{array}{l}A = {\left[ {\begin{array}{*{20}{c}}2&{ - 1}&3\end{array}} \right]_{1 \times 3}},k = 2,{\rm{ }}\\kA = 2A = 2\left[ {\begin{array}{*{20}{c}}2&{ - 1}&3\end{array}} \right]\end{array}\\\begin{array}{l} = \left[ {\begin{array}{*{20}{c}}{2(2)}&{2( - 1)}&{2(3)}\end{array}} \right]{\rm{ }}\\ = \left[ {\begin{array}{*{20}{c}}4&{ - 2}&6\end{array}} \right]\end{array}\end{array}\]
Solved Questions
1. Find the value of\[{\bf{A}} - {\bf{B}} + {\bf{2C}}\], where \[{\bf{A}} = \left[ {\begin{array}{*{20}{c}}1&2\end{array}} \right],{\rm{ }}{\bf{B}} = \left[ {\begin{array}{*{20}{c}}3&4\end{array}} \right],{\rm{ }}{\bf{C}} = \left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 2}\end{array}} \right]\].
Answer:
\[\begin{array}{*{20}{l}}{Let {\rm{ }}S = A - B + 2C}\\{ = \left[ {\begin{array}{*{20}{c}}1&2\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}3&4\end{array}} \right] + 2\left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 2}\end{array}} \right]}\\\begin{array}{l} = \left[ {\begin{array}{*{20}{c}}1&2\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}3&4\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{ - 2}&{ - 4}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{1 - 3 - 2}&{2 - 4 - 4}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ - 4}&{ - 6}\end{array}} \right]\end{array}\end{array}\]
2. Given\[{\bf{A}} = \left[ {\begin{array}{*{20}{c}}2&3&5\end{array}} \right],{\rm{ }}{\bf{B}} = \left[ {\begin{array}{*{20}{c}}{ - 1}&5&6\end{array}} \right],{\rm{ }}{\bf{M}} = \left[ {\begin{array}{*{20}{c}}{ - 4}&2&{ - 5}\end{array}} \right]\]. Find Matrix C for\[{\bf{M}} = {\bf{C}}+{\bf{3A}} + {\bf{2B}} \].
Answer:
Given \[{\bf{M}} = {\bf{3A}} + {\bf{2B}} +{\bf{C}}\]. Solving for C, we get \[C = M - 3A - 2B\]. Putting values of A, B, and M, we get
\[\begin{array}{*{20}{l}}{C = \left[ {\begin{array}{*{20}{c}}{ - 4}&2&{ - 5}\end{array}} \right] - 3\left[ {\begin{array}{*{20}{c}}2&3&5\end{array}} \right] - 2\left[ {\begin{array}{*{20}{c}}{ - 1}&5&6\end{array}} \right]}\\\begin{array}{l} = \left[ {\begin{array}{*{20}{c}}{ - 4}&2&{ - 5}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}6&9&{15}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}{ - 2}&{10}&{12}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ - 4 - 6 + 2}&{2 - 9 - 10}&{ - 5 - 15 - 12}\end{array}} \right]{\rm{ }}\\ = \left[ {\begin{array}{*{20}{c}}{ - 8}&{ - 17}&{ - 32}\end{array}} \right]\end{array}\end{array}\]
Practice Questions
1. If \[A = \left[ {\begin{array}{*{20}{c}}2&3&x\end{array}} \right]\], \[B = \left[ {\begin{array}{*{20}{c}}y&3&5\end{array}} \right]\] and A = B, then find the value of x and y.
Answer: x = 5, y = 2
2. If A and B two row matrices and AB exist, then find the number of columns of A.
Answer: The number of columns of A is 1.
3. Why does your Word document need to be formatted?
Conclusion
A Row Matrix is a horizontal matrix. It can also be called an array. Two or more Row Matrices can be added or subtracted if the order of both matrices is the same. We can multiply a scalar with a row matrix. Row matrix follows associativity property.
FAQs on Row Matrix: Definition, Properties & Examples
1. What is a row matrix and how is its order defined?
A row matrix is a type of matrix that consists of only a single row. The elements are arranged horizontally. Its order is always represented as 1 × n, where '1' signifies the single row and 'n' represents the number of columns. For example, the matrix A = [5 0 -2 9] is a row matrix of order 1 × 4.
2. What is the key difference between a row matrix and a column matrix?
The primary difference lies in their dimensions and orientation. A row matrix has exactly one row and multiple columns (order 1 × n), with its elements arranged horizontally. In contrast, a column matrix has exactly one column and multiple rows (order m × 1), with its elements arranged vertically.
3. Can you provide an example of a row matrix with 5 elements?
Yes. A row matrix with 5 elements will have 1 row and 5 columns, making its order 1 × 5. An example would be:
B = [12 -4 8 0 1].
Here, the matrix B contains a single row with five distinct elements.
4. What is the result when a row matrix of order 1 × n is multiplied by a column matrix of order n × 1?
When a row matrix of order 1 × n is multiplied by a column matrix of order n × 1, the resulting product is a 1 × 1 matrix. A 1 × 1 matrix contains only a single element and is often treated as a scalar. This operation is fundamental in linear algebra and is equivalent to calculating the dot product of two vectors.
5. How are row matrices related to the concept of vectors in linear algebra?
In the context of linear algebra, a row matrix is often referred to as a row vector. It can represent a point or a vector in an n-dimensional space, where 'n' is the number of columns. For instance, the row matrix [x y z] can represent a vector in a 3-dimensional Cartesian coordinate system. They are crucial for representing systems of linear equations and performing geometric transformations.
6. Is a single number, like 7, considered a row matrix?
No, a single number like 7 is a scalar, not a row matrix. A row matrix is a specific arrangement of elements in one row. However, the number 7 can be an element within a row matrix. For instance, in the matrix A = [7], 'A' is technically a 1 × 1 matrix, which has one row. But the term 'row matrix' is typically used to describe matrices of order 1 × n where n > 1 to distinguish them from 1 × 1 matrices or scalars.
7. What is the transpose of a row matrix?
The transpose of a row matrix is a column matrix. The process of transposing a matrix involves interchanging its rows and columns. Since a row matrix has an order of 1 × n, its transpose will have an order of n × 1, which is the definition of a column matrix. For example, if R = [2 4 6] is a row matrix, its transpose R' would be a column matrix:
[2]
[4]
[6]
