Question

Define Scalar Matrix.

Answer 1

Hint: We are asked to learn about the scalar matrix. For better understanding, we will first learn about the matrix and do some examples. Then we will learn about the square matrix. Then after this, we will learn about the diagonal matrix and will do certain examples to get a better understanding and lastly, we will learn about the scalar matrix. We will work on an example to check which one is scalar and which one is not.

Complete answer:
Before we define the scalar matrix, we should learn a little about the initial. First, we will understand what matrices are. In Mathematics, a set of numbers arranged in rows and columns so as to form a rectangular array. This is called a matrix. For example, \[\left[ \begin{matrix}
   2 \\
   1 \\
   1 \\
\end{matrix} \right],\left[ \begin{matrix}
   2 & 1 \\
   3 & 1 \\
\end{matrix} \right],\left[ \begin{matrix}
   1 & 2 & 3 \\
\end{matrix} \right].\] These are all examples of a matrix.
Secondly, we will learn what a square matrix is. A matrix in which the number of rows and columns are the same then those matrices are called a square matrix. For example, \[\left[ 1 \right],\left[ \begin{matrix}
   1 & 2 \\
   3 & 1 \\
\end{matrix} \right],\left[ \begin{matrix}
   1 & 2 & 3 \\
   4 & 1 & 1 \\
   5 & 6 & 7 \\
\end{matrix} \right].\]
Now, we will learn about the diagonal matrix. A square matrix in which the entries outside the main diagonal are all zero. The diagonal entries may or may not be zero, such matrices are called a diagonal matrix. Let us consider an example, \[\left[ \begin{matrix}
   1 & 0 \\
   0 & 2 \\
\end{matrix} \right],\] in this only diagonal entries are non – zero while non-diagonal entries are zero. So this is a diagonal matrix. In this example, \[\left[ \begin{matrix}
   0 & 0 \\
   0 & 0 \\
\end{matrix} \right],\] all non-diagonal entries are zero, so it is also called a diagonal matrix.
Now, finally, we will define the scalar matrix. A square matrix in which all the diagonal entries are equal and all non – diagonal entries are zero is called a scalar matrix. Or we can say a diagonal matrix in which all entries of diagonal are the same are called the scalar matrix. We will consider some examples.
\[\left[ \begin{matrix}
   1 & 2 \\
   2 & 1 \\
\end{matrix} \right]\]
Here the diagonal entries are the same but non – diagonal entries are not zero. So this is now a scalar matrix.
\[\left[ \begin{matrix}
   2 & 0 & 0 \\
   0 & 2 & 0 \\
   0 & 0 & 2 \\
\end{matrix} \right]\]
Here, the diagonal entries are 2 and non – diagonal entries are 0, so this is a scalar matrix.

Note:
Students should remember both the conditions need to be satisfied then only we will claim the matrix as a scalar matrix. The matrix entries can be any real number, it can be both positive as well as negative. Also, remember [2] is also a scalar matrix because the diagonal entry is 2 and no other entries are available. This is a scalar matrix of order 1.