Q.If for a square matrix $A$, $A. (\text{adj}A) = \begin{bmatrix} 2025 & 0 & 0 \\ 0 & 2025 & 0 \\ 0 & 0 & 2025 \end{bmatrix}$, then the value of $|A| + |\text{adj}A|$ is equal to:
(A) $1$
(B) $2025+1$
(C) $(2025)^2 + 45$
(D) $2025+(2025)^2$

Showing the 12 most recent of 20 on this concept.

1. 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 $(Adj\,A)_{22}=9\Rightarrow 3a-0=9\Rightarrow a=3$. …

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2. 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 $kA$: $k=|A|^3$. …

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3. 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 …

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4. 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. $|M|=9^3\cdot|P\cdot Adj\,Q|=729\cdot\dfrac19=81=9^2$. …

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5. AP EAPCET 2022Set eng-2022-07-08-FN1 markMCQ
   Q.If $A = \begin{pmatrix}1 & -3 & 2\\ -2 & 1 & 3\\ 3 & 2 & -1\end{pmatrix}$ then $A^2\, \mathrm{Adj}\, A =$ (A) $21A$ (B) $-42A$ (C) $7A^{-1}$ (D) $14(\mathrm{Adj}\,A)$

   ›Reveal solutionSolution

   This tests the identity $A\cdot\mathrm{Adj}(A) = (\det A)I$ applied cleverly to reduce $A^2\,\mathrm{Adj}\,A$.

   Concept and Intuition

   Rather than computing $\mathrm{Adj}\,A$ explicitly (tedious for a 3×3), use the fundamental relation $A\cdot\mathrm{Adj}\,A = (\det A)I$ to convert one factor of $A$ times $\mathrm{Adj}\,A$ directly into a scalar multiple of the identity, leaving a single $A$ behind.

   Step-by-Step Solution

   1. Compute $\det A$ for $A=\begin{pmatrix}1&-3&2\\-2&1&3\\3&2&-1\end{pmatrix}$: $\det A = 1(1\cdot(-1)-3\cdot2) -(-3)((-2)(-1)-3\cdot3) +2((-2)(2)-1\cdot3)$ $= 1(-7) +3(-7) +2(-7) = -7-21-14 = -42$. …

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6. AP EAPCET 2023Set eng-2023-05-18-FN1 markMCQ
   Q.If A and B are non-singular matrices and $\det(AB)=(\det A)(\det B)$, then $\big((\det A)(\det B)\big)B^{-1}A^{-1} =$ (A) $\mathrm{Adj}(BA)$ (B) $\mathrm{Adj}(A)+\mathrm{Adj}(B)$ (C) $\mathrm{Adj}(AB)$ (D) $(\mathrm{Adj}\,B)(\mathrm{Adj}\,A)$

   ›Reveal solutionSolution

   Using the identity det(M)M⁻¹ = Adj(M) with M = AB (noting (AB)⁻¹ = B⁻¹A⁻¹) gives the answer directly as Adj(AB).

   Concept and Intuition

   For any invertible square matrix M, the adjugate satisfies Adj(M) = det(M)·M⁻¹. This is a standard identity coming from M·Adj(M) = det(M)·I.

   Step-by-Step Solution

   1. Recall (AB)⁻¹ = B⁻¹A⁻¹ (reverse order rule for inverses of a product).
   2. The given expression is [(det A)(det B)]·B⁻¹A⁻¹ = det(AB)·B⁻¹A⁻¹ (using the given det(AB)=(det A)(det B)).
   3. Rewrite B⁻¹A⁻¹ as (AB)⁻¹.
   4. So the expression equals det(AB)·(AB)⁻¹. …

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7. AP EAPCET 2023Set eng-2023-05-15-AN1 markMCQ
   Q.If $\det(AB) = (\det A)(\det B)$ and $A$ is a non-singular matrix of order $3 \times 3$, then $\det(\text{adj } A) =$ (A) $\det(A)$ (B) $(\det(A))^{-1}$ (C) $(\det(A))^2$ (D) $(\det(A))^3$

   ›Reveal solutionSolution

   The standard identity $\det(\operatorname{adj} A) = (\det A)^{n-1}$ for an $n\times n$ matrix gives $(\det A)^2$ for $n=3$. Answer: (C).

   Concept and Intuition

   The adjugate satisfies $A \cdot \operatorname{adj}A = (\det A) I$. Taking determinants of both sides and using $\det(AB)=\det A \det B$ turns this matrix identity into a scalar one, letting us find $\det(\operatorname{adj}A)$ purely from $\det A$ and the matrix size $n$.

   Step-by-Step Solution

   1. Start from $A(\operatorname{adj}A) = (\det A) I_n$.
   2. Take determinants of both sides: $\det(A)\det(\operatorname{adj}A) = \det((\det A) I_n)$.
   3. For a scalar $k$ multiplying an $n\times n$ identity matrix, $\det(kI_n) = k^n$. Here $k=\det A$, so RHS is $(\det A)^n$.
   4. So $\det(A)\det(\operatorname{adj}A) = (\det A)^n$. …

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8. 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.

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

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

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9. 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 $B=\begin{bmatrix}1&2&3\\4&-3&-1\\2&1&-4\end{bmatrix}$: $\det B = 1((-3)(-4)-(-1)(1)) - 2(4(-4)-(-1)(2)) + 3(4(1)-(-3)(2))$ …

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10. AP EAPCET 2021Set eng-2021-10-05-FN1 markMCQ
    Q.If $A = \begin{bmatrix} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{bmatrix}$ then $adj(A) = ?$ (A) $2A^T$ (B) $A^T$ (C) $3A^T$ (D) $4A^T$

    ›Reveal solutionSolution

    $A$'s rows are mutually orthogonal with equal norm, so $AA^T=9I$; combined with $\det A=27$, this gives $\operatorname{adj}(A)=3A^T$ directly, without computing all nine cofactors.

    Concept and Intuition

    For any invertible square matrix, $\operatorname{adj}(A) = \det(A)\,A^{-1}$. If $A$ happens to have orthogonal rows of equal length (a scaled orthogonal matrix), $AA^T$ collapses to a scalar multiple of $I$, instantly giving $A^{-1}$ (and hence the adjugate) without a full cofactor expansion.

    Step-by-Step Solution

    1. $A=\begin{bmatrix}-1&-2&-2\\2&1&-2\\2&-2&1\end{bmatrix}$. Compute $\det A$ by cofactor expansion along row 1: $\det A = -1(1\cdot1-(-2)(-2)) -(-2)(2\cdot1-(-2)\cdot2) + (-2)(2(-2)-1\cdot2)$ $= -1(1-4) +2(2+4) -2(-4-2) = 3+12+12 = 27$. …

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11. 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): …

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12. 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 $|A| = \pm\dfrac{1}{4}$. …

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