Question:
If $A$ and $B$ are invertible matrices of same order, then which of the following is not correct?
If $A$ and $B$ are invertible matrices of same order, then which of the following is not correct?
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Always perform a "type check" on mathematical equations. The product of two matrices is a matrix. A determinant is a scalar. A matrix cannot equal a scalar. Spotting dimension/type mismatches is a quick way to identify false statements.
Updated On: Apr 29, 2026
- $A \cdot (\text{adj } A) = (\text{adj } A) \cdot A = |A|I$
- $A \cdot \text{adj } A = \text{adj } A \cdot A = |A|$
- $(AB)^{-1} = B^{-1} A^{-1}$
- $|A| \neq 0, |B| \neq 0$
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The Correct Option is B
Solution and Explanation
Step 1: Use standard matrix identities
Step 2: Check each option
(1) \[ A(\text{adj }A) = (\text{adj }A)A = |A|I \] Correct identity. (2) \[ A(\text{adj }A) = |A| \] Left side is a matrix, right side is a scalar $\Rightarrow$ dimension mismatch.
Hence incorrect. (3) \[ (AB)^{-1} = B^{-1}A^{-1} \] Correct property. (4) \[ |A| \neq 0,\ |B| \neq 0 \] True for invertible matrices.
Step 3: Conclusion
Only option (2) is incorrect. Final Answer:
\[ \boxed{\text{Option (2)}} \]
- $A \cdot (\text{adj } A) = |A|I$
- $(AB)^{-1} = B^{-1}A^{-1}$
- A matrix is invertible $\iff |A| \neq 0$
Step 2: Check each option
(1) \[ A(\text{adj }A) = (\text{adj }A)A = |A|I \] Correct identity. (2) \[ A(\text{adj }A) = |A| \] Left side is a matrix, right side is a scalar $\Rightarrow$ dimension mismatch.
Hence incorrect. (3) \[ (AB)^{-1} = B^{-1}A^{-1} \] Correct property. (4) \[ |A| \neq 0,\ |B| \neq 0 \] True for invertible matrices.
Step 3: Conclusion
Only option (2) is incorrect. Final Answer:
\[ \boxed{\text{Option (2)}} \]
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