Question

If I is a unit matrix, then 3I will be
(A) a unit matrix
(B) a triangular matrix
(C) a scalar matrix
(D) None of these

Answer 1

Hint: Assume a \[3\times 3\] matrix A whose diagonal elements are equal to 1 and the rest of the elements are equal to zero. Now, get the determinant value of I. We know that the unit matrix is a matrix that has a determinant value equal to 1. The diagonal elements of this matrix are equal to 1 and the rest of the element is equal to zero. So, A is equal to I. Now, get the matrix 3A. We also know that a scalar matrix is a matrix that has equal valued diagonal elements and all other remaining elements are equal to Zero. Now, conclude the answer about the matrix 3A.

Complete step-by-step solution -
According to the question, we have a unit matrix I and we have to find 3I.
We know that the unit matrix is a matrix that has a determinant value equal to 1. The diagonal elements of this matrix are equal to 1 and the rest of the element is equal to zero ………………….(1)
First of all, let us assume a square matrix A of order 3 i.e., \[3\times 3\] matrix. The matrix A has 1 as its elements only along its diagonal and its rest of the elements are equal to 0.
Now, we have our matrix A = \[\left[ \begin{align}
  & \begin{matrix}
   1 & 0 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 1 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 0 & 1 \\
\end{matrix} \\
\end{align} \right]\] …………………………..(2)
Calculating the determinant value of the matrix I,
\[\left| A \right|=\left| \begin{align}
  & \begin{matrix}
   1 & 0 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 1 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 0 & 1 \\
\end{matrix} \\
\end{align} \right|\]
\[\begin{align}
  & =1\left[ 1\left\{ 1\left( 1 \right)-0\left( 0 \right) \right\}-0\left\{ 0\left( 1 \right)-0\left( 0 \right) \right\}+0\left\{ 0\left( 0 \right)-0\left( 1 \right) \right\} \right] \\
 & =1\left[ 1\left( 1-0 \right)-0\left( 0-0 \right)+0\left( 0-0 \right) \right] \\
 & =1\left( 1 \right) \\
 & =1 \\
\end{align}\]
 We can see that the determinant value of matrix A is equal to 1 ……………………(3)
Now, from equation (1) and equation (3), we can say that our matrix A is the same as matrix I. It means matrix A is equal to matrix I.
\[\left[ A \right]=\left[ I \right]\] ………………………….(4)
Now, we need the value of 3I …………………(5)
From equation (4) and equation (5), we get
\[\left[ 3I \right]=\left[ 3A \right]\] …………………….(6)
Putting the value of matrix A in equation (6), we get
\[\left[ 3I \right]=\left[ 3A \right]=3\left[ \begin{align}
  & \begin{matrix}
   1 & 0 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 1 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 0 & 1 \\
\end{matrix} \\
\end{align} \right]\]
On multiplying, we get
\[\left[ 3I \right]=\left[ 3A \right]=\left[ \begin{align}
  & \begin{matrix}
   3 & 0 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 3 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 0 & 3 \\
\end{matrix} \\
\end{align} \right]\] ……………………(7)
We know that a scalar matrix is a matrix that has equal valued diagonal elements and all other remaining elements are equal to Zero.
Since the matrix 3I has equal valued diagonal elements and all other remaining elements equal to Zero so, we can say that 3I is a scalar matrix.
Therefore, 3I is a scalar matrix.
Hence, option (C) is the correct one.

Note: In this question, one might go with option (A) because in a unit matrix we also have equal valued diagonal elements and all other remaining elements equal to zero. This is wrong because the diagonal matrix should be equal to 1 and the matrix 3I has 3 as its diagonal elements. Therefore, the matrix 3I is not a unit matrix.