\( Q = \begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix} \) is a \( 2 \times 2 \) matrix.
Which one of the following statements is TRUE?
Show Hint
- \(Q\) is equal to its transpose.
- \(Q\) is equal to its inverse.
- \(Q\) is of full rank.
- \(Q\) has linearly dependent columns.
The Correct Option is C
Solution and Explanation
Step 1: Compute the determinant: \[ \det(Q) = 1\cdot1 - (-2)\cdot2 = 1+4 = 5 \neq 0. \] Hence \(Q\) is invertible.
Step 2: Since \(\det(Q) \neq 0\), the rank of \(Q\) is: \[ \text{rank}(Q) = 2 \quad (\text{full rank}). \] Therefore, the columns of \(Q\) are linearly independent.
Step 3: Check other properties: \[ Q \neq Q^{T}, \quad Q^{-1} = \tfrac{1}{5} \begin{bmatrix} 1 & 2 \\ -2 & 1 \end{bmatrix} \neq Q. \]
Therefore, the only correct statement is: \[ \boxed{\text{(C) is true.}} \]
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