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:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → B and D only
The question asks for the statements that are not true.
If B and C are both inverses of A, then
\( B=BI=B(AC)=(BA)C=IC=C \)
For a product, the order reverses:
\( (AB)(B^{-1}A^{-1})=A(BB^{-1})A^{-1} \)
\( (AB)(B^{-1}A^{-1})=AA^{-1}=I \)
Taking the transpose of \( AA^{-1}=I \):
\( (A^{-1})^T A^T=I \)
So the inverse is unique, \( (AB)^{-1}=B^{-1}A^{-1} \), \( (A^{-1})^{-1}=A \) and \( (A^T)^{-1}=(A^{-1})^T \).
Checking the statements
(A) Inverse of a matrix, if it exists, is unique. — Correct. Two inverses B and C of A must be equal.
(B) For two invertible matrices of same order, say \( A \) and \( B \), then \( (AB)^{-1}=A^{-1}B^{-1} \) — Incorrect. \( (AB)^{-1}=B^{-1}A^{-1} \), which in general differs from \( A^{-1}B^{-1} \) because matrix multiplication is not commutative.
(C) For an invertible matrix \( A \), \( (A^{-1})^{-1}=A \) — Correct. \( A^{-1}A=AA^{-1}=I \), so A is the inverse of \( A^{-1} \).
(D) For an invertible matrix \( A \), \( (A^T)^{-1}=A^T \) — Incorrect. \( (A^T)^{-1}=(A^{-1})^T \); for example, A = 2I gives \( (A^T)^{-1}=\frac{1}{2}I\ne 2I \).
So the statements that are not true are B and D.
CBSE Applied Mathematics (Class XII):
Inverse of a Matrix
