For a square matrix A, Which of the following is true?A. a) |A| = |AT|B. |A-1| = |A|C. |A-1| = |1A |D. Non

To determine the truth of the statements regarding a square matrix $A$, we will analyze each option.

1. Break Down the Problem

1. Evaluate statement A: $|A| = |A^T|$.
2. Evaluate statement B: $|A^{-1}| = |A|$.
3. Evaluate statement C: $|A^{-1}| = |1A|$.
4. Evaluate statement D: Non.

2. Relevant Concepts

• The determinant of a matrix $A$ is denoted by $|A|$.
• The determinant of the transpose of a matrix is equal to the determinant of the matrix itself: $|A| = |A^T|$.
• The determinant of the inverse of a matrix is the reciprocal of the determinant: $|A^{-1}| = \frac{1}{|A|}$.
• The determinant of the identity matrix $I_n$ (where $n$ is the dimension) is $|I_n| = 1$.

3. Analysis and Detail

1. Statement A: $|A| = |A^T|$

   • True. The determinant of a matrix is equal to the determinant of its transpose.

2. Statement B: $|A^{-1}| = |A|$

   • False. The correct relationship is $|A^{-1}| = \frac{1}{|A|}$.

3. Statement C: $|A^{-1}| = |1A|$

   • False. The notation $1A$ (which usually means scaling the matrix $A$ by 1) is just $A$, hence it simplifies to $|A^{-1}| = |A|$, which is false as mentioned above.

4. Statement D: Non

   • Since some statements are true, this option is false.

4. Verify and Summarize

• We verified that statement A is true.
• Statements B and C are false.
• Statement D is also false since at least one statement (A) is true.

Final Answer

The true statement is A: $|A| = |A^T|$.