Question

For a given square matrix \(A\) of order n, if there exists another square matrix \(B\) of the same order n, such that \(AB = BA = I\), then \(A\) is said to be invertible and \(B\) is called the inverse matrix of \(A\).

Which of the following statements are not TRUE ?

A. Inverse of a matrix, if it exists, is unique.
B. For two invertible matrices of same order, say \(A\) and \(B\) , then \((AB)^{-1} = A^{-1}B^{-1}\)
C. For an invertible matrix \(A\), \((A^{-1})^{-1} = A\)
D. For an invertible matrix \(A\), \((A^{T})^{-1} = A^{T}\)

Choose the correct answer from the options given below:

• A. A and C only
• B. B and D only
• C. A, C and D only
• D. A and D only

Hint:

Use \((AB)^{-1} = B^{-1}A^{-1}\) and \((A^T)^{-1} = (A^{-1})^T\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We must pick the statements about inverse matrices that are FALSE. So we test each statement against the standard properties of the inverse.

Step 2: Check statement (A).
Suppose \(B\) and \(C\) are both inverses of \(A\). Then \(B = BI = B(AC) = (BA)C = IC = C\). So the inverse is unique. So (A) is TRUE.

Step 3: Check statement (B).
The correct rule is the reversal law: \((AB)^{-1} = B^{-1}A^{-1}\). Matrix multiplication is not commutative, so \(A^{-1}B^{-1}\) is in general a different matrix. So (B) is FALSE.

Step 4: Check statement (C).
Since \(AA^{-1} = A^{-1}A = I\), the matrix \(A\) itself acts as the inverse of \(A^{-1}\). So \((A^{-1})^{-1} = A\), and (C) is TRUE.

Step 5: Check statement (D).
The correct rule is \((A^{T})^{-1} = (A^{-1})^{T}\). It equals \(A^{T}\) only in special cases, for example when \(A\) is orthogonal. For \(A = \begin{bmatrix}1 & 2\\3 & 5\end{bmatrix}\) the two sides are different. So (D) is FALSE.

Step 6: Pick the option.
The statements that are not true are B and D, which is option 2. Option 1 lists true statements. Options 3 and 4 wrongly include a true statement.

Final Answer:
The statements that are not true are B and D only. \[ \boxed{\text{B and D only}} \]