Question

Match List-I with List-II. List-I: A. If A is an invertible matrix then, B. If A is orthogonal matrix then, C. \((A-\lambda I)\) is a matrix, D. The rank of unit matrix of order n. List-II: I. \(|A|=\pm 1\), II. \(|A|\ne 0\), III. is \(n\), IV. Characteristic.

• A. A-I, B-II, C-III, D-IV
• B. A-II, B-I, C-IV, D-III
• C. A-II, B-I, C-III, D-IV
• D. A-I, B-II, C-IV, D-III

Hint:

Invertible matrix has non-zero determinant, orthogonal matrix has determinant \(\pm 1\), and rank of identity matrix of order \(n\) is \(n\).

Answer 1

The Correct Option is B

Concept:
This question is based on basic properties of matrices, determinant, orthogonal matrix, characteristic matrix, and rank of identity matrix.

Step 1: Match invertible matrix.
A matrix is invertible if its determinant is non-zero. \[ |A| \ne 0 \] Therefore, \[ A \rightarrow II \]

Step 2: Match orthogonal matrix.
For an orthogonal matrix \(A\): \[ A^T A = I \] Taking determinant on both sides: \[ |A^T A| = |I| \] \[ |A^T||A| = 1 \] \[ |A|^2 = 1 \] \[ |A| = \pm 1 \] Therefore, \[ B \rightarrow I \]

Step 3: Match \((A-\lambda I)\).
The matrix \((A-\lambda I)\) is called the characteristic matrix. \[ C \rightarrow IV \]

Step 4: Match rank of unit matrix.
The unit matrix of order \(n\) has \(n\) linearly independent rows and columns. Therefore, its rank is \(n\). \[ D \rightarrow III \] Therefore, the correct matching is: \[ A-II,\ B-I,\ C-IV,\ D-III \] \[ \therefore \text{Correct Answer is (B)} \]