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If A is a square matrix satisfying A’ A = I, write the value of |A|​

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CBSE

Mathematics

Grade 12

Determinants

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If A is a square matrix satisfying $\text{{A}'A}=\text{I}$ , write the value of $\left| \text{A} \right|$ .

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Hint: Firstly, we have to take the determinant on both the sides of the given equation, {A}'A=I . Then, we have to apply the properties of determinant mainly |AB|=|A||B| and |A|=∣∣A′∣∣ . We will use the property that the determinant of the identity matrix is always 1. Now, we have to simplify the resultant equation.

Complete step by step solution:

We are given that {A}'A=I . Let us take determinants on both sides.

⇒∣∣A′A∣∣=|I|

We know that if A and B are square matrix of same order, then |AB|=|A||B| . We also know that determinant of identity matrix is always 1, that is, |I|=1 .Therefore, we can write the above equation as

⇒∣∣A′∣∣|A|=1

We know that for any square matrix, A we can write |A|=∣∣A′∣∣ . Therefore, the above equation can be written as

⇒|A||A|=1

We can write the LHS as

⇒|A|2=1

Let us take square roots on both sides. We can write the result of this step as

⇒|A|=±1

Therefore, the value of ∣∣A∣∣ is ±1

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