The GATE Grind

GATE 2019 EE – Question 36

Engineering Mathematics · Linear Algebra: Matrix Algebra · 2 marks · Multiple choice

Consider a $2\times2$ matrix $M=[v_1\ \ v_2]$, where, $v_1$ and $v_2$ are the column vectors. Suppose $M^{-1}=\begin{bmatrix}u_1^T\\u_2^T\end{bmatrix}$, where $u_1^T$ and $u_2^T$ are the row vectors. Consider the following statements:

Statement 1: $u_1^Tv_1=1$ and $u_2^Tv_2=1$

Statement 2: $u_1^Tv_2=0$ and $u_2^Tv_1=0$

Which of the following options is correct?

  1. Statement 1 is true and statement 2 is false
  2. Statement 2 is true and statement 1 is false
  3. Both the statements are true
  4. Both the statements are false

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) Both the statements are true

Explanation

$M^{-1}M=I$, and the $(i,j)$ entry of the product is $u_i^Tv_j$. So $u_1^Tv_1=u_2^Tv_2=1$ and $u_1^Tv_2=u_2^Tv_1=0$. Both statements are true.