Question

If A is an orthogonal matrix, then value of \[|A|\] is equal to
a) \[ \pm 3\]
b) \[ \pm 4\]
c) \[ \pm 2\]
d) \[ \pm 1\]

Answer 1

Hint: Here the question is related to the topic matrix, since the matrix is an orthogonal matrix first we have to know the definition and condition of the orthogonal matrix. On considering the condition of the orthogonal matrix and the properties of the determinants we are determining the value of the determinant of matrix A.

Complete step by step answer:
In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns.
First we know the definition of orthogonal matrix.
If a matrix is an orthogonal matrix then \[A{A^T} = I\], this is a particular condition for the orthogonal matrix.
Now we will consider the given question
The A is an orthogonal matrix, then by considering the condition we have
\[ \Rightarrow A{A^T} = I\]
Here we have to determine the determinant of the matrix A
\[ \Rightarrow |A{A^T}| = |I|\]
By using the property of determinants \[|a.b| = |a|.|b|\], so we have
\[ \Rightarrow |A||{A^T}| = |I|\]
By the property \[|{A^T}| = |A|\], on substituting in the above equation we have
\[ \Rightarrow |A||A| = |I|\]
On multiplying we get
\[ \Rightarrow |A{|^2} = |I|\]
The value of the determinant of an identity matrix is 1
\[ \Rightarrow |A{|^2} = 1\]
On taking the square root both sides we get
\[ \Rightarrow |A| = \pm 1\]
If A is an orthogonal matrix, then value of \[|A|\] is equal to \[ \pm 1\].

So, the correct answer is “Option d”.

Note:
Here we are considering the determinant of an identity matrix is 1. So consider the \[2 \times 2\] identity matrix. \[|I| = \left| {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right|\], now we determine the value of the determinant of the identity matrix. \[|I| = 1 \times 1 - 0 \times 0\]. On simplifying we get \[|I| = 1\]. Whatever the order of a matrix, the determinant value of an identity matrix will be 1.