Question

From a set of $2\times 2$ matrices having 0 or 1 in each place, a matrix is chosen. The probability that it is a unit matrix is
A. $\dfrac{1}{16}$
B. $\dfrac{2}{16}$
C. $\dfrac{3}{16}$
D. $\dfrac{1}{4}$

Answer 1

Hint: To find the probability that the matrix chosen from $2\times 2$ matrices having 0 or 1 in each place is a unit matrix, we must first find in how many ways the digits can be placed inside a $2\times 2$ matrix. Since there are 2 possible values, that is, 0 and 1 and there are 4 positions in a $2\times 2$ matrix, the number of ways 0 and 1 are arranged is given as $2\times 2\times 2\times 2={{2}^{4}}=16\text{ ways}$ . There is only one possibility for a matrix to be a unit matrix. Hence, by substituting the values in the formula $P(A)=\dfrac{\text{Number of favorable outcome}}{\text{Total number of outcomes}}$ the required probability can be obtained.

Complete step-by-step solution
We need to find the probability that the matrix chosen from $2\times 2$ matrices having 0 or 1 in each place is a unit matrix.
Let us see in how many ways the digits can be placed inside a $2\times 2$ matrix. Consider a $2\times 2$ matrix shown below.
${{\left[ \begin{matrix}
   * & * \\
   * & * \\
\end{matrix} \right]}_{2\times 2}}$
The * positions can be filled by either 1 or 0. That is each position can get 2 values. Hence, we can find the number of ways in which 1 or 0 can be filled in a $2\times 2$ matrix by multiplying the possibilities, that is, 2 for each of the 4 positions. That is,
$2\times 2\times 2\times 2={{2}^{4}}=16\text{ ways}$
Now, we have to find the possibility of getting a unit matrix. Let us see what a unit matrix is.
A unit matrix is also known as an identity matrix. It has all the elements along the main diagonal as 1 and the remaining elements as 0. Let us see how it is denoted.
$\left[ \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right]$
The above matrix is a $2\times 2$ unit matrix. We can see that the diagonal elements are 1 and other elements are 0.
We know that a $2\times 2$ unit matrix can have only 1 unit matrix.
Let us now find the probability that the matrix is chosen as a unit matrix. We can denote this as $P(A)$. Hence,
$P(A)=\dfrac{\text{Number of favourable outcome}}{\text{Total number of outcomes}}$
We know that the number of favorable outcomes is 1 since a $2\times 2$ unit matrix can have only 1 unit matrix. We also found the total number of outcomes to be 16. Now, let us substitute these in the above formula. We will get
$P(A)=\dfrac{\text{1}}{\text{16}}$
Hence, the correct option is A.

Note: Do not get confused with the term ‘unit’ in unit matrix. You may think that a unit matrix is a matrix in which all elements are 1. This is a matrix of ones, not a unit matrix. You may make an error when writing the formula for probability. Do not write the probability formula as $P(A)=\dfrac{\text{Total number of outcomes}}{\text{Number of favourable outcome}}$