Q.If $A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$, then verify that $A\,\operatorname{adj} A = |A|\,I$. Also find $A^{-1}$.

Showing the 12 most recent of 20 on this concept.

1. AP EAPCET 2026Set eng-2026-05-15-FN1 markMCQ
   Q.For a $3 \times 3$ non singular matrix $A$, if $Adj(Adj(Adj(Adj(A)))) = |A|^n A$, then $n =$ (A) $3$ (B) $4$ (C) $8$ (D) $5$

   ›Reveal solutionSolution

   The key idea is that repeatedly applying the adjugate to a $3\times3$ matrix scales it by a power of its determinant. Using the property $\operatorname{Adj}(\operatorname{Adj}(A)) = |A|^{n-2} A$ for an $n\times n$ matrix, we find that after four adjugates the exponent is $3^4 = 81$, so $n = 81$ — but the problem’s given form $|A|^n A$ forces us to match exponents, leading to $n = 8$.

   We are given a $3 \times 3$ non-singular matrix $A$ and the equation

   $\operatorname{Adj}(\operatorname{Adj}(\operatorname{Adj}(\operatorname{Adj}(A)))) = |A|^n A.$

   We need to find $n$.

   ---

   1. Recall the fundamental adjugate property

   For any invertible $m \times m$ matrix $M$,

   $\operatorname{Adj}(M) = |M| \cdot M^{-1}.$

   This is the definition: the adjugate is the transpose of the cofactor matrix, and it satisfies $M \cdot \operatorname{Adj}(M) = |M| I$.

   2. Apply it once

   Let $A$ be $3 \times 3$. Then

   $\operatorname{Adj}(A) = |A| \cdot A^{-1}.$

   3. Apply it twice

   Now compute $\operatorname{Adj}(\operatorname{Adj}(A))$.

   Let $B = \operatorname{Adj}(A) = |A| A^{-1}$.

   Then

   $\operatorname{Adj}(B) = |B| \cdot B^{-1}.$

   We need $|B|$:

   $|B| = \bigl| |A| A^{-1} \bigr| = |A|^3 \cdot |A^{-1}| = |A|^3 \cdot \frac{1}{|A|} = |A|^2.$

   Also $B^{-1} = (|A| A^{-1})^{-1} = \frac{1}{|A|} A$.

   Thus

   $\operatorname{Adj}(\operatorname{Adj}(A)) = |A|^2 \cdot \frac{1}{|A|} A = |A| \, A.$

   Tip

   For an $m \times m$ matrix, $\operatorname{Adj}(\operatorname{Adj}(A)) = |A|^{m-2} A$. Here $m=3$, so $|A|^{3-2} = |A|^1$, matching our result.

   4. Apply it three times

   Let $C = \operatorname{Adj}(\operatorname{Adj}(A)) = |A| A$.

   Then

   $\operatorname{Adj}(C) = |C| \cdot C^{-1}.$

   Now $|C| = \bigl| |A| A \bigr| = |A|^3 \cdot |A| = |A|^4$.

   And $C^{-1} = (|A| A)^{-1} = \frac{1}{|A|} A^{-1}$.

   So

   $\operatorname{Adj}(C) = |A|^4 \cdot \frac{1}{|A|} A^{-1} = |A|^3 A^{-1}.$

   5. Apply it four times

   Let $D = \operatorname{Adj}(\operatorname{Adj}(\operatorname{Adj}(A))) = |A|^3 A^{-1}$.

   Then

   $\operatorname{Adj}(D) = |D| \cdot D^{-1}.$

   Compute $|D| = \bigl| |A|^3 A^{-1} \bigr| = (|A|^3)^3 \cdot |A^{-1}| = |A|^9 \cdot \frac{1}{|A|} = |A|^8$.

   And $D^{-1} = (|A|^3 A^{-1})^{-1} = \frac{1}{|A|^3} A$.

   Thus

   $\operatorname{Adj}(D) = |A|^8 \cdot \frac{1}{|A|^3} A = |A|^5 A.$

   Watch out

   A common mistake is to think the exponent grows linearly. Actually, each adjugate multiplies the exponent by $m-1$ and adds something — here it compounds quickly: $1 \to 2 \to 4 \to 8$ for the determinant factor, but the matrix factor alternates between $A$ and $A^{-1}$.

   6. Compare with the given form

   We have found

   $\operatorname{Adj}^4(A) = |A|^5 A.$

   The problem states this equals $|A|^n A$. Therefore $n = 5$.

   ✓Final answer

   The correct option is (D).

   ANSWER: D

2. AP EAPCET 2026Set eng-2026-05-18-FN1 markMCQ
   Q.If $A = \begin{bmatrix} 2 & -3 & 1 \\ -3 & 1 & -2 \\ 1 & -2 & 3 \end{bmatrix}$, then $Adj(A) =$ (A) $\begin{bmatrix} 1 & -7 & 5 \\ -7 & 5 & -1 \\ 5 & -1 & 7 \end{bmatrix}$ (B) $\begin{bmatrix} 1 & 7 & -5 \\ 7 & 5 & 1 \\ -5 & 1 & 7 \end{bmatrix}$ (C) $\begin{bmatrix} -1 & 7 & 5 \\ 7 & 5 & 1 \\ 5 & 1 & -7 \end{bmatrix}$ (D) $\begin{bmatrix} -1 & 7 & -5 \\ 7 & -5 & 1 \\ -5 & 1 & -7 \end{bmatrix}$

   ›Reveal solutionSolution

   $A$ is symmetric, so its adjoint equals its cofactor matrix; computing the cofactors gives $\begin{bmatrix}-1&7&5\\7&5&1\\5&1&-7\end{bmatrix}$.

   Setup. $\operatorname{Adj}(A)$ is the transpose of the cofactor matrix. Here

   $A=\begin{bmatrix}2 & -3 & 1\\-3 & 1 & -2\\1 & -2 & 3\end{bmatrix}$

   is symmetric, so the adjoint is also symmetric.

   Cofactors.

   $C_{11}=\begin{vmatrix}1&-2\\-2&3\end{vmatrix}=3-4=-1,\quad C_{12}=-\begin{vmatrix}-3&-2\\1&3\end{vmatrix}=-(-9+2)=7,\quad C_{13}=\begin{vmatrix}-3&1\\1&-2\end{vmatrix}=6-1=5,$

   $C_{22}=\begin{vmatrix}2&1\\1&3\end{vmatrix}=6-1=5,\quad C_{23}=-\begin{vmatrix}2&-3\\1&-2\end{vmatrix}=-(-4+3)=1,\quad C_{33}=\begin{vmatrix}2&-3\\-3&1\end{vmatrix}=2-9=-7.$

   By symmetry $C_{21}=7,\;C_{31}=5,\;C_{32}=1$.

   Adjoint. Transposing the cofactor matrix (which is symmetric here) gives

   $\operatorname{Adj}(A)=\begin{bmatrix}-1 & 7 & 5\\7 & 5 & 1\\5 & 1 & -7\end{bmatrix}.$

   ✓Final answer

   $\operatorname{Adj}(A)=\begin{bmatrix}-1&7&5\\7&5&1\\5&1&-7\end{bmatrix}$ — option (C).

3. AP EAPCET 2026Set eng-2026-05-18-AN1 markMCQ
   Q.$A, B$ are $3^{rd}$ order non-singular square matrices and $K$ is a real number. Which of the following is true? (A) $Adj(AB) = (Adj\, B)(Adj\, A)$ and $adj(A^{-1}) \neq (adj\, A)^{-1}$ (B) $Adj(KA) = K\, Adj(A)$ and $|KA| = K^3|A|$ (C) $|B^{-1}AB| = |A|$ and $(A+B)^2 = A^2 + 2AB + B^2$ (D) $(adj\, A)^{-1} = \dfrac{A}{|A|}$ and $(AB)^{-1} = B^{-1}A^{-1}$

   ›Reveal solutionSolution

   The key idea is to test each statement using standard matrix properties for non‑singular matrices. Only option (D) contains two statements that are both true.

   1. Check option (A):

      • The first part, $\operatorname{Adj}(AB) = (\operatorname{Adj} B)(\operatorname{Adj} A)$, is a true property of adjugates for square matrices (order doesn’t matter as long as they are square).
      • The second part claims $\operatorname{Adj}(A^{-1}) \neq (\operatorname{Adj} A)^{-1}$. But we know $\operatorname{Adj}(A^{-1}) = |A^{-1}| A = \frac{1}{|A|} A$, and $(\operatorname{Adj} A)^{-1} = \frac{A}{|A|}$ (since $\operatorname{Adj} A = |A| A^{-1}$). These are equal, so the inequality is false. Hence (A) is not fully true.

   2. Check option (B):

      • $\operatorname{Adj}(KA) = K^{n-1} \operatorname{Adj} A$ for an $n \times n$ matrix. Here $n=3$, so $\operatorname{Adj}(KA) = K^{2} \operatorname{Adj} A$, not $K \operatorname{Adj} A$. So the first part is false.
      • The second part $|KA| = K^3 |A|$ is true (since determinant scales by $K^n$). But because the first part is false, (B) is incorrect.

   3. Check option (C):

      • $|B^{-1}AB| = |B^{-1}| |A| |B| = \frac{1}{|B|} |A| |B| = |A|$ — this is true (similarity transformation preserves determinant).
      • However, $(A+B)^2 = A^2 + AB + BA + B^2$. For matrices, $AB \neq BA$ in general, so $(A+B)^2 = A^2 + 2AB + B^2$ holds only if $A$ and $B$ commute. No such condition is given, so this is false. Hence (C) is not fully true.

   4. Check option (D):

      • First part: $(\operatorname{Adj} A)^{-1} = \frac{A}{|A|}$. This is a standard result: since $\operatorname{Adj} A = |A| A^{-1}$, taking inverse gives $(\operatorname{Adj} A)^{-1} = \frac{A}{|A|}$. True.
      • Second part: $(AB)^{-1} = B^{-1}A^{-1}$ is the fundamental reversal law for inverses of products. True.
      • Both statements are correct, so (D) is the correct option.

   Watch out

   A common mistake is to assume $\operatorname{Adj}(KA) = K \operatorname{Adj} A$; the correct factor is $K^{n-1}$.

   Tip

   For adjugates, remember $\operatorname{Adj}(A) = |A| A^{-1}$ for invertible matrices — this lets you quickly verify many properties.

   ✓Final answer

   The correct option is (D).

   ANSWER: D

4. AP EAPCET 2025Set eng-2025-05-21-AN1 markMCQ
   Q.If $A=\begin{pmatrix}a & b & c\\ d & e & f\\ l & m & n\end{pmatrix}$ is a matrix such that $|A|>0$ and $\mathrm{Adj}\,A=\begin{pmatrix}0 & 4 & -6\\ 10 & 8 & 0\\ 2 & 4 & -4\end{pmatrix}$, then $\dfrac{cd}{fb}+\dfrac{ln}{em}=$ (A) $2a$ (B) $a+m$ (C) $a+b$ (D) $a$

   ›Reveal solutionSolution

   Reconstructing $A$ from $\mathrm{adj}(A)=A^{-1}\det(A)$ via $A=\mathrm{adj}(\mathrm{adj}\,A)/\det(A)$ gives concrete entries, letting $\frac{cd}{fb}+\frac{log}{em}$ be evaluated directly and matched to $a+m$.

   Concept and Intuition

   Since $A\cdot\mathrm{adj}(A)=\det(A)\cdot I$, once $\mathrm{adj}(A)$ is fully known and $\det(A)$ is pinned down, $A$ itself is uniquely determined: $A=\det(A)\cdot[\mathrm{adj}(A)]^{-1}=\dfrac{\mathrm{adj}(\mathrm{adj}\,A)}{\det(A)}$ (using $\mathrm{adj}(\mathrm{adj}\,A)=\det(A)^{n-2}A$ for $n=3$). This turns an abstract cofactor question into a concrete numeric matrix, from which any algebraic combination of entries can just be computed.

   Step-by-Step Solution

   1. $\det(\mathrm{adj}\,A)=\det(A)^{2}$ for a $3\times3$ matrix. Computing $\det\begin{pmatrix}0&4&-6\\10&8&0\\2&4&-4\end{pmatrix}=16$, so $\det(A)^2=16\Rightarrow\det(A)=\pm4$; given $|A|>0$, $\det(A)=4$.
   2. Compute $\mathrm{adj}(\mathrm{adj}\,A)$ (cofactor-transpose of the given adjugate matrix): $\begin{pmatrix}-32&-8&48\\40&12&-60\\24&8&-40\end{pmatrix}$.
   3. $A=\dfrac{\mathrm{adj}(\mathrm{adj}\,A)}{\det A}=\dfrac14\begin{pmatrix}-32&-8&48\\40&12&-60\\24&8&-40\end{pmatrix}=\begin{pmatrix}-8&-2&12\\10&3&-15\\6&2&-10\end{pmatrix}$.
   4. Read off: $a=-8,\,b=-2,\,c=12,\,d=10,\,e=3,\,f=-15,\,l=6,\,m=2,\,n=-10$.
   5. Verify consistency: $cd=120=af=(-8)(-15)=120$ correct; $en=-30=fm=(-15)(2)=-30$ correct; and $\det(A)$ recomputes to $4$ correct.
   6. $\dfrac{cd}{fb}=\dfrac{120}{(-15)(-2)}=\dfrac{120}{30}=4$; $\dfrac{log}{em}=\dfrac{(6)(-10)}{(3)(2)}=\dfrac{-60}{6}=-10$.
   7. Sum $=4-10=-6$. Compare to options: $a+m=-8+2=-6$ — matches.

   Common Mistakes

   • Trying to guess the answer symbolically without reconstructing $A$ — the combination $\frac{cd}{fb}+\frac{log}{em}$ doesn't simplify to a clean symbolic identity without knowing actual entry values here.
   • Sign errors in the cofactor-of-cofactor computation (easy to mismatch which row/column to delete).

   ✓Final answer

   The correct option is (B) — $a+m$.

   ANSWER: B

5. AP EAPCET 2025Set eng-2025-05-22-AN1 markMCQ
   Q.If $A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 5 \\ 2 & 1 & 6 \end{bmatrix}$ and $|\mathrm{adj}(\mathrm{adj}\,A)|(\mathrm{adj}\,A)^{-1} = kA$, then $k =$ (A) 1296 (B) 216 (C) 36 (D) 432

   ›Reveal solutionSolution

   Using the standard adjugate identities for a $3\times3$ matrix, $k=|A|^3$; direct computation gives $|A|=6$, so $k=216$ — option (B).

   Concept and Intuition

   The adjugate (classical adjoint) of an $n\times n$ matrix satisfies two workhorse identities: $|\mathrm{adj}\,A| = |A|^{n-1}$, and (applying that twice) $|\mathrm{adj}(\mathrm{adj}\,A)| = |A|^{(n-1)^2}$. Also, since $A\cdot\mathrm{adj}\,A = |A|\,I$, we get $\mathrm{adj}\,A = |A|A^{-1}$, i.e. $(\mathrm{adj}\,A)^{-1} = \dfrac{A}{|A|}$. Combining these turns the whole expression into a scalar power of $|A|$ times $A$ — exactly the form $kA$ the question wants.

   Step-by-Step Solution

   1. For $n=3$: $|\mathrm{adj}\,A|=|A|^{n-1}=|A|^2$, so $|\mathrm{adj}(\mathrm{adj}\,A)| = |\mathrm{adj}\,A|^{n-1} = (|A|^2)^{2} = |A|^4$.
   2. $(\mathrm{adj}\,A)^{-1} = \dfrac{A}{|A|}$ (from $A\,\mathrm{adj}\,A=|A|I$).
   3. So $|\mathrm{adj}(\mathrm{adj}\,A)|\,(\mathrm{adj}\,A)^{-1} = |A|^4\cdot\dfrac{A}{|A|} = |A|^3 A$. Comparing to $kA$: $k=|A|^3$.
   4. Compute $|A|$ for $A=\begin{bmatrix}1&2&3\\1&3&5\\2&1&6\end{bmatrix}$: expand along row 1: $1(3\cdot6-5\cdot1)-2(1\cdot6-5\cdot2)+3(1\cdot1-3\cdot2) = 1(18-5)-2(6-10)+3(1-6) = 13+8-15=6$.
   5. So $k=|A|^3=6^3=216$.

   Common Mistakes

   • Using the wrong exponent for $|\mathrm{adj}(\mathrm{adj}\,A)|$ — it's $(n-1)^2$, not $n-1$ or $(n-1)(n-2)$; double-apply the rule carefully.
   • Arithmetic slips in the $3\times3$ determinant expansion — recheck cofactor signs.

   ✓Final answer

   The correct option is (B) — 216.

   ANSWER: B

6. AP EAPCET 2025Set eng-2025-05-23-AN1 markMCQ
   Q.If $P=\begin{bmatrix}1 & \alpha & 3\\ 1 & 3 & 3\\ 2 & 4 & 4\end{bmatrix}$ is the adjoint of a matrix A and $\det A = 4$, then the value of $\alpha$ is (A) 3 (B) 22 (C) 11 (D) 4

   ›Reveal solutionSolution

   Uses $\det(\operatorname{adj}A)=(\det A)^{n-1}$ for an $n\times n$ matrix; with $n=3,\det A=4$, we need $\det P=16$, which pins down $\alpha=11$.

   Concept and Intuition

   The adjoint (adjugate) matrix of $A$ satisfies $A\cdot\operatorname{adj}A=(\det A)I$, and taking determinants of both sides gives the identity $\det(\operatorname{adj}A)=(\det A)^{n-1}$ for an $n\times n$ matrix. For $n=3$ this is $(\det A)^2$, always non-negative regardless of the sign of $\det A$ — so knowing $\det A=4$ immediately tells us $\det(\operatorname{adj}A)=16$ without needing $A$ itself.

   Step-by-Step Solution

   1. $P=\operatorname{adj}A$ is $3\times3$, so $\det P=(\det A)^{3-1}=(\det A)^2=4^2=16$.
   2. Expand $\det P$ along row 1:

   $\det P = 1\begin{vmatrix}3&3\\4&4\end{vmatrix}-\alpha\begin{vmatrix}1&3\\2&4\end{vmatrix}+3\begin{vmatrix}1&3\\2&4\end{vmatrix}$

   3. Compute the $2\times2$ determinants: $\begin{vmatrix}3&3\\4&4\end{vmatrix}=12-12=0$; $\begin{vmatrix}1&3\\2&4\end{vmatrix}=4-6=-2$.
   4. So $\det P = 1(0)-\alpha(-2)+3(-2) = 2\alpha-6$.
   5. Set $2\alpha-6=16 \implies 2\alpha=22 \implies \alpha=11$.

   Common Mistakes

   • Using $\det(\operatorname{adj}A)=\det A$ (wrong power) instead of $(\det A)^{n-1}$.
   • Sign errors in cofactor expansion, especially the alternating $+,-,+$ pattern along a row.

   ✓Final answer

   The correct option is (C) — $11$.

   ANSWER: C

7. AP EAPCET 2025Set eng-2025-05-24-FN1 markMCQ
   Q.If $a$ is the determinant of the adjoint of the matrix $\begin{bmatrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 3 \end{bmatrix}$ and $b$ is the determinant of the inverse of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 4 & -3 & -1 \\ 2 & 1 & -4 \end{bmatrix}$ then $\dfrac{b+1}{18b} =$ (A) $a$ (B) $10a$ (C) $2+a$ (D) $2a$

   ›Reveal solutionSolution

   This tests the identities $\det(\operatorname{adj}A)=(\det A)^{n-1}$ and $\det(A^{-1})=1/\det A$, then simplifying an algebraic expression in $b$ to match $a$.

   Concept and Intuition

   For an $n\times n$ matrix, $\det(\operatorname{adj}A) = (\det A)^{n-1}$; for $n=3$ this is $(\det A)^2$, always non-negative. Also, since $A\cdot A^{-1}=I$, taking determinants gives $\det(A^{-1}) = 1/\det A$. Both facts let us compute $a$ and $b$ purely from the determinants of the given matrices, then plug into the target expression.

   Step-by-Step Solution

   1. Compute $\det A$ for $A=\begin{bmatrix}1&1&2\\1&2&3\\2&3&3\end{bmatrix}$: $\det A = 1(2\cdot3-3\cdot3) - 1(1\cdot3-3\cdot2) + 2(1\cdot3-2\cdot2) = 1(-3)-1(-3)+2(-1) = -3+3-2=-2$.
   2. $a = \det(\operatorname{adj}A) = (\det A)^{2} = (-2)^2 = 4$.
   3. Compute $\det B$ for $B=\begin{bmatrix}1&2&3\\4&-3&-1\\2&1&-4\end{bmatrix}$: $\det B = 1((-3)(-4)-(-1)(1)) - 2(4(-4)-(-1)(2)) + 3(4(1)-(-3)(2))$ $= 1(12+1) - 2(-16+2) + 3(4+6) = 13 + 28 + 30 = 71$.
   4. $b = \det(B^{-1}) = \dfrac{1}{71}$.
   5. Now evaluate $\dfrac{b+1}{18b} = \dfrac{\frac{1}{71}+1}{18\cdot\frac1{71}} = \dfrac{\frac{72}{71}}{\frac{18}{71}} = \dfrac{72}{18} = 4$.
   6. Since $a=4$, we get $\dfrac{b+1}{18b} = a$.

   Common Mistakes

   • Using $\det(\operatorname{adj}A)=\det A$ instead of $(\det A)^{n-1}$.
   • Sign errors in the $3\times3$ cofactor expansion.

   ✓Final answer

   The correct option is (A) — $a$.

   ANSWER: A

8. AP EAPCET 2025Set eng-2025-05-26-AN1 markMCQ
   Q.If B is the inverse of a third order matrix A and $\det B = k$, then $(\mathrm{adj}\,(\mathrm{adj}\, A))^{-1} =$ (A) $kB$ (B) $\dfrac{1}{k}B$ (C) $kB^{-1}$ (D) $B + kI$

   ›Reveal solutionSolution

   Using $\operatorname{adj}(\operatorname{adj} A) = (\det A)^{n-2}A$ for $n=3$, together with $\det A = 1/k$, gives $(\operatorname{adj}(\operatorname{adj} A))^{-1}=kB$.

   Concept and Intuition

   For an $n\times n$ invertible matrix, the double-adjugate identity is $\operatorname{adj}(\operatorname{adj} A) = (\det A)^{n-2}A$. For $n=3$ this simplifies neatly to $(\det A)\,A$ — a single power of the determinant times $A$ itself. Combining this with $B=A^{-1}$ (so $\det B = 1/\det A$) lets everything be expressed back in terms of $B$ and $k$.

   Step-by-Step Solution

   1. $B = A^{-1}$ and $\det B = k$ $\Rightarrow$ $\det A = \dfrac{1}{\det B} = \dfrac1k$.
   2. For a $3\times3$ matrix: $\operatorname{adj}(\operatorname{adj} A) = (\det A)^{3-2}A = (\det A)A$.
   3. Take the inverse of both sides:

   $(\operatorname{adj}(\operatorname{adj} A))^{-1} = \left[(\det A)A\right]^{-1} = \frac{1}{\det A}A^{-1} = \frac{1}{\det A}B$

   4. Substitute $\dfrac{1}{\det A} = k$ (from step 1):

   $(\operatorname{adj}(\operatorname{adj} A))^{-1} = kB$

   Common Mistakes

   • Using the general-$n$ exponent incorrectly (forgetting it reduces to exactly $(\det A)^1$ for $n=3$, not $(\det A)^2$ as some students recall from the single-adjugate formula $\det(\operatorname{adj} A)=(\det A)^{n-1}$).
   • Mixing up whether $\det A = k$ or $\det A = 1/k$ — remember $B$ is the inverse of $A$, so their determinants are reciprocals.

   ✓Final answer

   The correct option is (A) — $kB$.

   ANSWER: A

9. AP EAPCET 2024Set eng-2024-05-21-AN1 markMCQ
   Q.Assertion (A): If $B$ is a $3 \times 3$ matrix and $|B| = 6$, then $|Adj(B)| = 36$. Reason (R): If $B$ is a square matrix of order $n$, then $|Adj(B)| = |B|^n$ (A) Both (A) and (R) are true and (R) is the correct explanation of (A) (B) Both (A) and (R) are true but (R) is not the correct explanation of (A) (C) (A) is true but (R) is false (D) (A) is false but (R) is true

   ›Reveal solutionSolution

   The correct identity is $|\mathrm{Adj}(B)|=|B|^{n-1}$; this makes Assertion (A)'s numeric value (36) actually correct, but Reason (R) states the wrong general formula ($|B|^n$), so (R) is false even though (A) is true.

   Concept and Intuition

   For an $n\times n$ invertible matrix $B$, the adjugate satisfies $B\cdot\mathrm{Adj}(B) = |B| I$. Taking determinants of both sides: $|B|\cdot|\mathrm{Adj}(B)| = |B|^n$ (since $\det(|B|I) = |B|^n$ for an $n\times n$ identity scaled by $|B|$). Dividing by $|B|$ (nonzero, since $B$ is invertible as $|B|=6\ne0$) gives $|\mathrm{Adj}(B)| = |B|^{n-1}$.

   Step-by-Step Solution

   1. Derive the correct formula: from $B\cdot\mathrm{Adj}(B)=|B|I_n$, take determinants: $|B|\cdot|\mathrm{Adj}(B)| = |B|^n$, so $|\mathrm{Adj}(B)|=|B|^{n-1}$ (for invertible $B$).
   2. Apply to this problem: $n=3$, $|B|=6$. Correct value: $|\mathrm{Adj}(B)| = 6^{3-1}=6^2=36$.
   3. Assertion (A) claims $|\mathrm{Adj}(B)|=36$ — this numeric value is correct (matches the true formula's output), so (A) is true.
   4. Reason (R) states the general rule as $|\mathrm{Adj}(B)|=|B|^n$ — but the actual rule is $|B|^{n-1}$. As a general statement, (R) is false (if you actually used $|B|^n$ here you'd wrongly get $6^3=216\ne36$).
   5. So (A) is true, but the "reason" given for it is a false formula — even though it happens to have been misapplied in a way that this specific example's numbers don't immediately expose (since the correct exponent for this problem, 2, isn't directly visible unless you check the formula itself).

   Common Mistakes

   • Assuming that because (A)'s stated number (36) is correct, (R) — which is offered as its justification — must also be correct. Always independently verify the general formula in (R), not just whether the final number in (A) checks out.
   • Mixing up $|B|^{n-1}$ with $|B|^n$ — a very common error; the exponent is always (order $-1$), not the order itself.

   ✓Final answer

   The correct option is (C) — (A) is true but (R) is false.

   ANSWER: C

10. AP EAPCET 2024Set eng-2024-05-21-FN1 markMCQ
    Q.Let $A$ be a $4 \times 4$ matrix and $P$ be its adjoint matrix. If $|P| = \left|\dfrac{A}{2}\right|$, then $|A^{-1}| =$ (A) $\pm \dfrac{1}{4}$ (B) $\pm 8$ (C) $\pm 2$ (D) $\pm 4$

    ›Reveal solutionSolution

    Using |adj(A)| = |A|^(n−1) for n=4 and |kA| = k^n|A|, the given equation reduces to |A|² = 1/16, so |A| = ±1/4 and |A⁻¹| = ±4.

    Concept and Intuition

    Two standard determinant identities are needed here:

    • For an $n\times n$ matrix $A$, $\det(\text{adj}\,A) = (\det A)^{n-1}$ (this follows from $A\cdot\text{adj}(A) = |A|I$, taking determinants of both sides: $|A|\cdot|\text{adj}\,A| = |A|^n$, so if $|A|\neq0$, $|\text{adj}\,A|=|A|^{n-1}$).
    • For a scalar $k$ and $n\times n$ matrix $A$: $\det(kA) = k^n\det(A)$ (each of the $n$ rows contributes a factor of $k$).

    Step-by-Step Solution

    1. $A$ is $4\times4$, so $n=4$. Given $P=\text{adj}(A)$, we have $|P| = |A|^{4-1} = |A|^3$.
    2. $\left|\dfrac{A}{2}\right| = \left(\dfrac12\right)^4|A| = \dfrac{1}{16}|A|$.
    3. Given $|P| = \left|\dfrac{A}{2}\right|$: $|A|^3 = \dfrac{1}{16}|A|$.
    4. Since $A^{-1}$ is asked for, $A$ must be invertible, i.e. $|A|\neq 0$. Divide both sides by $|A|$ (valid since $|A|\neq0$): $|A|^2 = \dfrac{1}{16}$.
    5. So $|A| = \pm\dfrac{1}{4}$.
    6. $|A^{-1}| = \dfrac{1}{|A|} = \dfrac{1}{\pm1/4} = \pm4$.

    Common Mistakes

    • Using $|\text{adj}\,A|=|A|^n$ instead of the correct $|A|^{n-1}$.
    • Forgetting the $\left(\dfrac12\right)^4$ scaling factor (using $n=4$, not some other power) when computing $|A/2|$.
    • Dividing by $|A|$ without first confirming $|A|\neq0$ (though here it's guaranteed since $A^{-1}$ is asked for).

    ✓Final answer

    The correct option is (D) — $\pm 4$.

    ANSWER: D

11. AP EAPCET 2024Set eng-2024-05-22-FN1 markMCQ
    Q.If $P$ and $Q$ are two $3 \times 3$ matrices such that $|PQ| = 1$ and $|P| = 9$, then the determinant of adjoint of the matrix $P \cdot Adj\, 3Q$ is (A) $9^4$ (B) $\dfrac{1}{9^4}$ (C) $9^2$ (D) $\dfrac{1}{9^2}$

    ›Reveal solutionSolution

    Using $|Q|=1/|P|\cdot|PQ|$, the scaling property $Adj(kA)=k^{n-1}AdjA$, and $|Adj\,M|=|M|^{n-1}$ for $3\times3$ matrices gives $9^4$.

    Concept and Intuition

    For an $n\times n$ matrix, $|kA|=k^n|A|$, $Adj(kA)=k^{n-1}Adj\,A$, and $|Adj\,A|=|A|^{n-1}$. Chaining these scaling rules for $n=3$ solves the problem without ever computing $P$ or $Q$ explicitly.

    Step-by-Step Solution

    1. $|PQ|=|P||Q|=1\Rightarrow |Q|=1/9$ (since $|P|=9$).
    2. $Adj(3Q)=3^{3-1}\,Adj\,Q=9\,Adj\,Q$.
    3. Let $M=P\cdot Adj(3Q)=9\,(P\cdot Adj\,Q)$.
    4. $|P\cdot Adj\,Q|=|P|\cdot|Adj\,Q|=|P|\cdot|Q|^{3-1}=9\cdot\left(\dfrac19\right)^2=\dfrac{9}{81}=\dfrac19$.
    5. $|M|=9^3\cdot|P\cdot Adj\,Q|=729\cdot\dfrac19=81=9^2$.
    6. $|Adj\,M|=|M|^{3-1}=|M|^2=(9^2)^2=9^4$.

    Common Mistakes

    • Forgetting the $k^{n-1}$ scaling factor when pulling the scalar $3$ out of $Adj(3Q)$.
    • Forgetting to cube (scale by $9^3$) when pulling the scalar $9$ out of the $3\times3$ matrix $M$'s determinant.

    ✓Final answer

    The correct option is (A) — $9^4$.

    ANSWER: A

12. AP EAPCET 2024Set eng-2024-05-22-FN1 markMCQ
    Q.If $A = \begin{bmatrix} a & 1 & 2 \\ 1 & 2 & b \\ c & 1 & 3 \end{bmatrix}$ and $Adj\, A = \begin{bmatrix} 7 & -1 & -5 \\ -3 & 9 & 5 \\ 1 & -3 & 5 \end{bmatrix}$ then $a^2 + b^2 + c^2 =$ (A) $10$ (B) $14$ (C) $11$ (D) $29$

    ›Reveal solutionSolution

    Each entry of $Adj\,A$ is the corresponding cofactor of $A$, transposed: $(Adj\,A)_{ij}=C_{ji}$. Writing out these equations pins down $a,b,c$ uniquely. Answer: $a^2+b^2+c^2=10$.

    Concept and Intuition

    $Adj\,A$ is the transpose of the cofactor matrix of $A$. So $(Adj\,A)_{ij}=C_{ji}(A)$ where $C_{ji}$ is the cofactor obtained by deleting row $j$ and column $i$ of $A$. Comparing entries of the given $Adj\,A$ to cofactors computed from the unknown entries $a,b,c$ of $A$ gives a solvable (over-determined but consistent) system.

    Step-by-Step Solution

    1. With $A=\begin{bmatrix}a&1&2\\1&2&b\\c&1&3\end{bmatrix}$, compute cofactor $C_{11}=\begin{vmatrix}2&b\\1&3\end{vmatrix}=6-b$. This equals $(Adj\,A)_{11}=7\Rightarrow b=-1$.
    2. Cofactor $C_{31}=\begin{vmatrix}1&2\\2&b\end{vmatrix}=b-4$. This equals $(Adj\,A)_{13}=-5\Rightarrow b-4=-5\Rightarrow b=-1$ (consistent).
    3. Cofactor $C_{12}=-\begin{vmatrix}1&b\\c&3\end{vmatrix}=bc-3$, equal to $(Adj\,A)_{21}=-3\Rightarrow bc=0$. With $b=-1$, get $c=0$.
    4. Cofactor $C_{22}=\begin{vmatrix}a&2\\c&3\end{vmatrix}=3a-2c$, equal to $(Adj\,A)_{22}=9\Rightarrow 3a-0=9\Rightarrow a=3$.
    5. Check consistency: $C_{33}=\begin{vmatrix}a&1\\1&2\end{vmatrix}=2a-1=5\Rightarrow a=3$ ✓; $C_{32}=-\begin{vmatrix}a&2\\1&b\end{vmatrix}=2-ab=2-3(-1)=5=(Adj\,A)_{23}$ ✓.
    6. So $a=3,b=-1,c=0\Rightarrow a^2+b^2+c^2=9+1+0=10$.

    Common Mistakes

    • Mixing up $(Adj\,A)_{ij}$ with the un-transposed cofactor $C_{ij}$ instead of $C_{ji}$.
    • Solving only one equation and not cross-checking with the others (the system is over-determined and must be fully consistent).

    ✓Final answer

    The correct option is (A) — $10$.

    ANSWER: A