Question

How do I find the additive identity matrix?

Answer 1

Hint: In this type of question, we should have a proper knowledge in matrix and determinant topic. The question is asked to find the additive identity matrix of any matrix. First, we will know what additive identity matrix is. After that, we will get to know how to find the additive identity matrix.

Complete step by step answer:
Let us solve this question.
Let us first know what the additive identity of a matrix is.
Additive identity states that when zero is added to any real number, that number remains the same.
Let X be any real number.
x+0=x
Here, 0 is the additive identity.
Similarly, an additive identity matrix states that when a zero matrix of any order is added to a matrix A (say) of the same order, then the addition will be same as matrix A. It means that zero matrix is not affecting the matrix A by adding the matrix A and matrix 0. Let there be a matrix 0 where each element of a matrix is zero.
Matrix A + Matrix 0 = Matrix A
\[\Rightarrow \left[ \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right]+\left[ \begin{matrix}
   0 & 0 & 0 \\
   0 & 0 & 0 \\
   0 & 0 & 0 \\
\end{matrix} \right]=\left[ \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right]\]
Or,
Matrix 0 + Matrix A = Matrix A
\[\Rightarrow \left[ \begin{matrix}
   0 & 0 & 0 \\
   0 & 0 & 0 \\
   0 & 0 & 0 \\
\end{matrix} \right]+\left[ \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right]=\left[ \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right]\]
This was for the order of the \[3\times 3\] matrix.
For any of the order of the matrices like \[1\times 1\], \[2\times 2\], \[1\times 2\], \[2\times 1\], \[3\times 1\], \[1\times 3\], \[n\times n\], \[m\times n\], (where m and n natural numbers) and many more will not get affected if we add them with zero matrix(where all elements are zero). Here, zero matrix is the additive identity matrix.
So, whenever we have to find the additive identity matrix, then first we will check the order of the given matrix A(say). After that, we will make a matrix 0(say) of the same order having all elements zero in that matrix 0.
That matrix 0 will be an additive identity matrix when added to A and after the addition of those matrices, we get the same matrix A.

Note: Let us see how to check if a matrix is an additive identity or not.
Let us have a matrix \[A=\left[ \begin{matrix}
   1 & 5 \\
   9 & 7 \\
\end{matrix} \right]\] and an additive identity matrix \[0=\left[ \begin{matrix}
   0 & 0 \\
   0 & 0 \\
\end{matrix} \right]\].
\[A+0=\left[ \begin{matrix}
   1 & 5 \\
   9 & 7 \\
\end{matrix} \right]+\left[ \begin{matrix}
   0 & 0 \\
   0 & 0 \\
\end{matrix} \right]=\left[ \begin{matrix}
   1+0 & 5+0 \\
   9+0 & 7+0 \\
\end{matrix} \right]=\left[ \begin{matrix}
   1 & 5 \\
   9 & 7 \\
\end{matrix} \right]=A\]
The condition is satisfied. This means that matrix 0 is the additive identity matrix of A.
Now, let us check if 0 matrix is added to the matrix A then what will we get
\[0+A=\left[ \begin{matrix}
   0 & 0 \\
   0 & 0 \\
\end{matrix} \right]+\left[ \begin{matrix}
   1 & 5 \\
   9 & 7 \\
\end{matrix} \right]=\left[ \begin{matrix}
   0+1 & 0+5 \\
   0+9 & 0+7 \\
\end{matrix} \right]=\left[ \begin{matrix}
   1 & 5 \\
   9 & 7 \\
\end{matrix} \right]=A\]
It is clear that a zero matrix or additive identity matrix has not changed the value of matrix A.
So, the additive identity of a matrix is the matrix of the same order having all the elements as zero.