Algebra

Matrices & Determinants

Medium

core

If A and B are two invertible matrices, then which of the following statements are correct?

(A) $|A^{-1}| = |A|^{-1}$

(B) $adjA = |A|A^{-1}$

(C) $(AB)^{-1} = A^{-1}B^{-1}$

(D) $(A + B)^{-1} = A^{-1} + B^{-1}$

Choose the correct answer from the options given below:

(A), (B) and (C) only

(A) and (B) only

(A), (B) and (D) only

✅ Correct Option: 2

For $|A^{-1}| = |A|^{-1}$ :

Since $A \cdot A^{-1} = I$ , taking determinants:

$|A \cdot A^{-1}| = |I|$

$|A| \cdot |A^{-1}| = 1$

$|A^{-1}| = \frac{1}{|A|}$

$|A^{-1}| = |A|^{-1}$

Statement (A) is correct.

For $adjA = |A|A^{-1}$ :

The fundamental relation is $A \cdot (adjA) = |A| \cdot I$

Multiplying both sides by $A^{-1}$ from the left:

$A^{-1} \cdot A \cdot (adjA) = A^{-1} \cdot |A| \cdot I$

$I \cdot (adjA) = |A| \cdot A^{-1}$

$adjA = |A|A^{-1}$

Statement (B) is correct.

For $(AB)^{-1} = A^{-1}B^{-1}$ :

The correct formula is $(AB)^{-1} = B^{-1}A^{-1}$ (reverse order).

Checking: $(AB) \cdot (B^{-1}A^{-1}) = A \cdot (B \cdot B^{-1}) \cdot A^{-1}$

$= A \cdot I \cdot A^{-1}$

$= A \cdot A^{-1}$

$= I$

However, $(AB) \cdot (A^{-1}B^{-1}) = A \cdot (B \cdot A^{-1}) \cdot B^{-1}$ , which does not simplify to $I$ since $B$ and $A^{-1}$ do not commute.

Statement (C) is incorrect.

For $(A + B)^{-1} = A^{-1} + B^{-1}$ :

Taking $A = I$ and $B = I$ :

$A + B = 2I$

$(A + B)^{-1} = (2I)^{-1} = \frac{1}{2}I$

But $A^{-1} + B^{-1} = I + I = 2I$

Since $\frac{1}{2}I \neq 2I$ , the formula is incorrect.

Statement (D) is incorrect.

Only statements (A) and (B) are correct.

2022: 16 July Shift 2

Read the following statements carefully:

A. Determinant is a square matrix

B. If A be any given square matrix of order n, then $A(\text{adj}A) = (\text{adj}A)A = |A|I$ .

C. If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.

D. If A is a nonsingular matrix, then its inverse does not exist

Which of the above statements are true?

Choose the correct answer from the options given below:

2025: 13 May Shift 2

Which of the following statements is (are) true?

(A) $B^T AB$ is a skew-symmetric matrix if A is a symmetric matrix

(B) $B^T AB$ is a symmetric matrix if A is a symmetric matrix

(D) $B^T AB$ is a skew-symmetric matrix if B is a skew-symmetric matrix

(E) $B^T AB$ is a symmetric matrix if B is a symmetric matrix

Choose the correct answer from the options given below:

2026: 25 May Shift 1

If A is a $2 \times 3$ matrix and B is a matrix such that $A^{T}B$ and $BA^{T}$ are both defined. Then order of B is

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If $\begin{vmatrix}6 & 7\\ 3 & 8\end{vmatrix} = \begin{vmatrix}3x & 6\\ -1 & x\end{vmatrix}$ , then the sum of values of $x$ is equal

2023: 15 June Shift 2

If A is a square matrix of order 3 such that $|A|=2$ , then the value of $|adj(adj A)|$ is :

2026: 30 May Shift 2

If $A = \begin{bmatrix}0 & a & 1\\-1 & b & 1\\-1 & c & 0\end{bmatrix}$ is a skew symmetric matrix, then the value of $(a+b-c)^2$ is

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