Matrices & Determinants
Transpose of a matrix
Grade Class 12
Question:
If A is a non-singular matrix and A<sup>T</sup> denotes the transpose of A, then :
(A) |A| ≠ |A<sup>T</sup>|
(B) |A. A<sup>T</sup>| = |A|<sup>2</sup>
(C) |A<sup>T</sup>. A| = |A<sup>T</sup>|<sup>2</sup>
(D) |A| + |A<sup>T</sup>| ≠ 0
Step-by-Step Solution
Key Concept: The determinant of a matrix is equal to the determinant of its transpose, i.e., |A| = |A^T|. Also, |AB| = |A||B|.
Since |A| = |A<sup>T</sup>|, we have |A.A<sup>T</sup>| = |A||A<sup>T</sup>| = |A||A| = |A|<sup>2</sup>. Thus, option (B) is correct.
Correct Answer: 2