Question

Pick out the wrong statement. If \(A\) and \(B\) are square matrices of the same order, then

• A. \(A + B = B + A\)
• B. \((AB)' = B'A'\)
• C. \(A - B = I\)
• D. \(|AB| = |A||B|\)

Hint:

Only identities that hold for all matrices are universally true.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question tests fundamental properties of matrix algebra, including commutativity of addition, transpose of product, and determinant of product.
Step 2: Detailed Explanation:
Let's analyze each statement:
(A) Statement A: Matrix addition is always commutative. For any two matrices \( A \) and \( B \) of same order, \( A + B = B + A \). This is a True statement.
(B) Statement B: The transpose of the product of two matrices is the product of their transposes in reverse order. This is a fundamental property. This is a True statement.
(C) Statement C: \( A - B = I \). This implies that the difference between any two square matrices of the same order must always result in the Identity matrix. This is clearly False, as the difference depends entirely on the elements of \( A \) and \( B \).
(D) Statement D: The determinant of the product of two square matrices is equal to the product of their individual determinants. This is a True property.
Step 3: Final Answer:
The wrong statement is \( A - B = I \).