CUET UG 2025 — Mathematics Algebra

For invertible matrices A and B, examining each statement:

Option 1: adj A = |A|A⁻¹

The relationship between inverse and adjoint is:

$$A^{-1} = \frac{\text{adj A}}{|A|}$$A−1=∣A∣adj A​

Multiplying both sides by |A|:

$$\text{adj A} = |A|A^{-1}$$adj A=∣A∣A−1

This statement is correct.

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Option 2: $(A + B)^{-1} = A^{-1} + B^{-1}$(A+B)−1=A−1+B−1

The inverse does not distribute over addition.

Counterexample using $A = [2]$A=[2] and $B = [3]$B=[3]:

$$A + B = [5]$$A+B=[5]

$$(A + B)^{-1} = [1/5] = 0.2$$(A+B)−1=[1/5]=0.2

However:

$$A^{-1} + B^{-1} = [1/2] + [1/3] = 0.5 + 0.333... = 0.833...$$A−1+B−1=[1/2]+[1/3]=0.5+0.333...=0.833...

Since $0.2 \neq 0.833$0.2=0.833, this statement is NOT correct.

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Option 3: $|A^{-1}| = |A|^{-1}$∣A−1∣=∣A∣−1

From the identity $A \times A^{-1} = I$A×A−1=I:

$$|A \times A^{-1}| = |I|$$∣A×A−1∣=∣I∣

$$|A| \times |A^{-1}| = 1$$∣A∣×∣A−1∣=1

$$|A^{-1}| = \frac{1}{|A|} = |A|^{-1}$$∣A−1∣=∣A∣1​=∣A∣−1

This statement is correct.

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Option 4: $(AB)^{-1} = B^{-1}A^{-1}$(AB)−1=B−1A−1

Verification:

$$(AB) \times (B^{-1}A^{-1}) = A(BB^{-1})A^{-1}$$(AB)×(B−1A−1)=A(BB−1)A−1

$$= AIA^{-1}$$=AIA−1

$$= AA^{-1}$$=AA−1

$$= I$$=I

This statement is correct.

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Therefore, Option 2 is NOT correct.