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 19 on this concept.

1. COMEDK 2026Set 2026-A1 markMCQ
   Q.Matrix $A=\left[\begin{array}{ccc}1 & 1 & 2 \\ 1 & -2 & 2 \\ 1 & 0 & -1\end{array}\right]$, Given $\boldsymbol{M}_{\mathbf{2 2}}$ and $\boldsymbol{A}_{\mathbf{3 2}}$ are the minor and cofactor of the adjoint matrix of $\boldsymbol{A}$ respectively then the value of the expression $\boldsymbol{M}_{\mathbf{2 2}}+\boldsymbol{A}_{\mathbf{3 2}}-|\boldsymbol{a} \boldsymbol{d} \boldsymbol{j}|$ is: (A) $-729$ (B) $-117$ (C) $-81$ (D) $-99$

   ›Reveal solutionSolution

   $\det A=9\Rightarrow|\operatorname{adj}A|=9^2=81$. On the adjoint matrix, $M_{22}=-18$ and $A_{32}=18$, so $M_{22}+A_{32}-|\operatorname{adj}A|=-18+18-81=-81$. The correct option is (C).

   Concept

   The minor and cofactor referred to are those of the adjoint matrix, so the adjoint must be built explicitly. Also, for an $n\times n$ matrix $|\operatorname{adj}A|=(\det A)^{n-1}$, which here gives $|\operatorname{adj}A|=(\det A)^2$.

   Solution

   1. Determinant. $\det A=1(2)-1(-3)+2(2)=9$.
   2. $|\operatorname{adj}A|$. $(\det A)^2=81$.
   3. Adjoint. Transposing the cofactor matrix of $A$,

   $\operatorname{adj}A=\begin{bmatrix}2&1&6\\3&-3&0\\2&1&-3\end{bmatrix}.$

   4. Minor $M_{22}$. Delete row 2, column 2: $\begin{vmatrix}2&6\\2&-3\end{vmatrix}=-6-12=-18.$
   5. Cofactor $A_{32}$. Delete row 3, column 2 and attach $(-1)^{3+2}=-1$: $-\begin{vmatrix}2&6\\3&0\end{vmatrix}=-(0-18)=18.$
   6. Combine. $M_{22}+A_{32}-|\operatorname{adj}A|=-18+18-81=-81.$

   ✓Final answer

   The value is $-81$ — option (C).

2. COMEDK 2026Set 2026-M1 markMCQ
   Q.Given $P=\left[\begin{array}{lll}2 & \boldsymbol{\alpha} & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 3\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix A and $|A|=3$, then the value of $\boldsymbol{\alpha}$ is: (A) 7 (B) -25 (C) -8 (D) -26

   ›Reveal solutionSolution

   The key idea is that for any square matrix A, we have $A \cdot \text{adj}(A) = |A| I$. Given $P = \text{adj}(A)$ and $|A| = 3$, we can multiply A by P and equate to $3I$ to solve for $\alpha$. The value is $\alpha = -8$, so option (C) is correct.

   ---

   We start with the fundamental relationship between a matrix and its adjoint. For any $3 \times 3$ matrix $A$,

   $A \cdot \text{adj}(A) = |A| I$

   where $I$ is the identity matrix. Here we are told $P = \text{adj}(A)$ and $|A| = 3$, so

   $A \cdot P = 3I.$

   This means $P$ is essentially the inverse of $A$ scaled by 3. But we don't know $A$ directly — we only know $P$. However, we can use the fact that the determinant of $P$ is related to $|A|$:

   $|P| = |\text{adj}(A)| = |A|^{n-1} = 3^{2} = 9.$

   That gives us an equation in $\alpha$ directly, without needing $A$ at all.

   ---

   1. Compute $|P|$ in terms of $\alpha$.

   $P = \begin{pmatrix} 2 & \alpha & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 3 \end{pmatrix}$

   Expand along the first row:

   $|P| = 2 \cdot \begin{vmatrix} 2 & 2 \\ 3 & 3 \end{vmatrix} - \alpha \cdot \begin{vmatrix} 1 & 2 \\ 1 & 3 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 2 \\ 1 & 3 \end{vmatrix}.$

   Compute each minor:

   • $\begin{vmatrix} 2 & 2 \\ 3 & 3 \end{vmatrix} = (2)(3) - (2)(3) = 0$.
   • $\begin{vmatrix} 1 & 2 \\ 1 & 3 \end{vmatrix} = (1)(3) - (2)(1) = 1$.

   So:

   $|P| = 2 \cdot 0 - \alpha \cdot 1 + 1 \cdot 1 = -\alpha + 1.$

   2. Use the determinant relation. We know $|P| = |\text{adj}(A)| = |A|^{2} = 3^2 = 9$. Therefore:

   $-\alpha + 1 = 9 \quad \Rightarrow \quad -\alpha = 8 \quad \Rightarrow \quad \alpha = -8.$

   Tip

   A common pitfall is to try to find $A$ first by inverting $P$, but that’s unnecessary. The determinant relation $|\text{adj}(A)| = |A|^{n-1}$ is the cleanest shortcut here.

   Watch out

   Don’t forget the sign pattern when expanding a determinant — the second term gets a minus sign because of the checkerboard pattern. Missing that would give $\alpha = 10$, which is not among the options.

   ---

   ✓Final answer

   The correct option is (C).

   ANSWER: C

3. KCET 2026Set UNKNOWN1 markMCQ
   Q.If $A$ and $B$ are invertible matrices of same order, then which of the following is not correct? (A) $A\cdot(\text{adj}\,A) = (\text{adj}\,A)\cdot A = |A|I$ (B) $A\cdot \text{adj}\,A = \text{adj}\,A\cdot A = |A|$ (C) $(AB)^{-1} = B^{-1}A^{-1}$ (D) $|A| \neq 0, |B| \neq 0$

   ›Reveal solutionSolution

   Check each identity against the standard results for invertible matrices; the one missing the identity matrix $I$ on the right side is incorrect.

   Step 1 — Check option (A)

   For any square matrix $A$, the standard identity is $A \cdot (\text{adj}\,A) = (\text{adj}\,A)\cdot A = |A|\,I$, where $I$ is the identity matrix of the same order. This is a correct, standard result.

   Step 2 — Check option (B)

   Option (B) states $A\cdot \text{adj}\,A = \text{adj}\,A \cdot A = |A|$. The right-hand side here is just the scalar $|A|$, not $|A|I$. But $A\cdot(\text{adj}\,A)$ is a matrix (of the same order as $A$), and a matrix can never equal a bare scalar unless trivially $1\times 1$. This statement is incorrect as written — it drops the identity matrix $I$.

   Step 3 — Check options (C) and (D)

   $(AB)^{-1} = B^{-1}A^{-1}$ is the standard reversal rule for inverses of a product — correct. Since $A,B$ are given as invertible, $|A|\neq 0$ and $|B|\neq 0$ is simply the definition of invertibility — correct.

   ✓Final answer

   The correct option is (B) — $A\cdot \text{adj}\,A = \text{adj}\,A\cdot A = |A|$ is NOT correct (it should equal $|A|I$).

4. COMEDK 2025Set 2025-A1 markMCQ
   Q.If $A(\operatorname{adj} A)=\left[\begin{array}{lll}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{array}\right]$, then the value of $|A|+|\operatorname{adj} A|$ is equal to : (A) 5 (B) 25 (C) 125 (D) 30

   ›Reveal solutionSolution

   The key idea is that for any square matrix $A$, $A(\operatorname{adj} A) = |A| I$. Here that gives $|A| = 5$, and since $|\operatorname{adj} A| = |A|^{n-1}$ for an $n \times n$ matrix, we get $|\operatorname{adj} A| = 5^2 = 25$. Their sum is $5 + 25 = 30$, so the answer is (D).

   ---

   The problem gives us $A(\operatorname{adj} A) = 5I$, where $I$ is the $3 \times 3$ identity matrix. This is a classic property: for any square matrix $A$, the product $A$ times its adjugate equals the determinant times the identity. So the scalar on the diagonal is exactly $|A|$. That means $|A| = 5$ immediately.

   Now, we also need $|\operatorname{adj} A|$. There's a neat formula: for an $n \times n$ matrix, $|\operatorname{adj} A| = |A|^{n-1}$. Here $n = 3$, so $|\operatorname{adj} A| = 5^{2} = 25$.

   Thus the sum is $5 + 25 = 30$.

   Let's walk through it step by step.

   1. Recall the defining property of the adjugate. For any square matrix $A$, we have

   $A (\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I.$

   This is the fundamental relation. The problem gives us the left-hand side explicitly as $5I$, so we can directly compare:

   $|A| I = 5I \quad \Rightarrow \quad |A| = 5.$

   2. Find $|\operatorname{adj} A|$ using the determinant of both sides. Take determinants of the equation $A (\operatorname{adj} A) = |A| I$:

   $|A| \cdot |\operatorname{adj} A| = \big| |A| I \big|.$

   The right-hand side is a scalar matrix: $|A| I$ is $|A|$ times the identity, so its determinant is $(|A|)^n$ for an $n \times n$ matrix. Here $n = 3$, so

   $\big| |A| I \big| = (|A|)^3.$

   Thus

   $|A| \cdot |\operatorname{adj} A| = (|A|)^3.$

   Since $|A| \neq 0$ (it's 5), we can divide both sides by $|A|$:

   $|\operatorname{adj} A| = (|A|)^2 = 5^2 = 25.$

   3. Add them up.

   $|A| + |\operatorname{adj} A| = 5 + 25 = 30.$

   Tip

   A common shortcut: for a $3 \times 3$ matrix, $|\operatorname{adj} A| = |A|^2$ always holds. So once you spot $|A| = 5$, the sum is $5 + 25 = 30$ without any further work.

   Watch out

   A classic mistake is to think $|\operatorname{adj} A| = |A|$ or to forget the exponent $n-1$. Always check the size: for $2 \times 2$, it's $|A|^1$; for $3 \times 3$, it's $|A|^2$; for $n \times n$, it's $|A|^{n-1}$.

   ✓Final answer

   The correct option is (D).

   ANSWER: D

5. COMEDK 2025Set 2025-E1 markMCQ
   Q.Kiran purchased 3 pencils, 2 notebooks and one pen for ₹41. From the same shop Manasa purchased 2 pencils, one notebook and 2 pens for ₹ 29 , while Shreya purchased 3 pencils, 2 notebooks and 2 pens for ₹ 44. The above situation can be represented in matrix form as $A X=B$. Then $|\operatorname{adj} A|$ is equal to (A) 9 (B) $-$9 (C) $-$1 (D) 1

   ›Reveal solutionSolution

   The coefficient matrix has $\det A=-1$, and for a $3\times3$ matrix $|\operatorname{adj} A|=|A|^{2}=1$ — option (D).

   Set up the system. Let pencil $=x$, notebook $=y$, pen $=z$:

   $3x+2y+z=41,\quad 2x+y+2z=29,\quad 3x+2y+2z=44.$

   So

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

   Determinant (expand along the first row):

   $\det A = 3(1\cdot2-2\cdot2) - 2(2\cdot2-2\cdot3) + 1(2\cdot2-1\cdot3)$

   $= 3(-2) - 2(-2) + 1(1) = -6+4+1 = -1.$

   Adjugate determinant. For an $n\times n$ matrix, $|\operatorname{adj} A| = |A|^{\,n-1}$. Here $n=3$:

   $|\operatorname{adj} A| = |A|^{2} = (-1)^{2} = 1.$

   ✓Final answer

   $|\operatorname{adj} A| = 1$ — (D) $1$

6. COMEDK 2025Set 2025-E1 markMCQ
   Q.Value of the determinant of a matrix $A$ of order $3 \times 3$ is 7 . Then the value of the determinant formed by the cofactors of matrix A is (A) 7 (B) 49 (C) 14 (D) 343

   ›Reveal solutionSolution

   The determinant of the cofactor matrix (the adjugate) of a $3\times3$ matrix $A$ equals $(\det A)^{n-1} = (\det A)^2$. Given $\det A = 7$, the answer is $7^2 = 49$, so option (B).

   ---

   The key idea is that the matrix of cofactors is intimately linked to the inverse of $A$. For any square matrix $A$, the product $A \cdot (\text{adj } A) = (\det A) I$, where $\text{adj } A$ is the transpose of the cofactor matrix. Taking determinants on both sides gives a direct relationship between $\det(\text{cofactor matrix})$ and $\det A$.

   For an $n \times n$ matrix, the determinant of the cofactor matrix (strictly, of the adjugate) is $(\det A)^{n-1}$. Here $n=3$, so the exponent is $2$.

   Let’s walk through it carefully.

   1. Define the cofactor matrix and adjugate. For a $3 \times 3$ matrix $A$, let $C_{ij}$ be the cofactor of entry $a_{ij}$. The cofactor matrix is $C = [C_{ij}]$. The adjugate (or classical adjoint) is the transpose: $\operatorname{adj}(A) = C^T$. The fundamental property is:

   $A \cdot \operatorname{adj}(A) = \operatorname{adj}(A) \cdot A = (\det A) I_3.$

   This holds for any square matrix.

   2. Take determinants of both sides. From $A \cdot \operatorname{adj}(A) = (\det A) I_3$, we have:

   $\det(A \cdot \operatorname{adj}(A)) = \det((\det A) I_3).$

   The left side, by the product rule, is $\det A \cdot \det(\operatorname{adj}(A))$.

   The right side: multiplying a $3\times3$ identity matrix by the scalar $\det A$ gives a diagonal matrix with $\det A$ on each diagonal entry. Its determinant is $(\det A)^3$.

   3. Set up the equation.

   $\det A \cdot \det(\operatorname{adj}(A)) = (\det A)^3.$

   Since $\det A = 7 \neq 0$, we can divide both sides by $\det A$:

   $\det(\operatorname{adj}(A)) = (\det A)^2 = 7^2 = 49.$

   4. Relate to the cofactor matrix. The determinant of the cofactor matrix $C$ is the same as the determinant of its transpose $\operatorname{adj}(A)$, because $\det(C^T) = \det(C)$. So:

   $\det(\text{cofactor matrix}) = \det(\operatorname{adj}(A)) = 49.$

   Tip

   For an $n \times n$ matrix, the pattern is $\det(\text{adj}(A)) = (\det A)^{n-1}$. Here $n=3$ gives exponent 2. For $2\times2$ it would be exponent 1 (so just $\det A$), and for $4\times4$ exponent 3, etc.

   Watch out

   A common mistake is to think the determinant of the cofactor matrix equals $\det A$ itself. That’s only true for $2\times2$ matrices. For $3\times3$, you must square it.

   ✓Final answer

   The correct option is (B).

   ANSWER: B

7. COMEDK 2025Set 2025-M1 markMCQ
   Q.The sum of three numbers is 6 . Twice the third number, when added to the first number gives 7 , On adding the sum of the second and third numbers to thrice the first number, we get 12 . The above situation can be represented in matrix form as $A X=B$. Then the $|\operatorname{adj} A|$ is equal to (A) $-$4 (B) 4 (C) $-$64 (D) 16

   ›Reveal solutionSolution

   We translate the word problem into a system of three linear equations, write it in matrix form $AX = B$, compute the determinant of $A$, and then use the property $|\operatorname{adj} A| = |A|^{n-1}$ with $n=3$ to get the answer $|\operatorname{adj} A| = 16$.

   ---

   We start by turning the story into equations.

   Let the three numbers be $x$, $y$, and $z$.

   1. Translate the conditions

      • “The sum of three numbers is 6” → $x + y + z = 6$.
      • “Twice the third number, when added to the first number gives 7” → $x + 2z = 7$.
      • “On adding the sum of the second and third numbers to thrice the first number, we get 12” → $3x + (y + z) = 12$, i.e. $3x + y + z = 12$.

      So the system is:

   $\begin{cases} x + y + z = 6 \\ x + 0y + 2z = 7 \\ 3x + y + z = 12 \end{cases}$

   2. Write in matrix form $AX = B$

   $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ 3 & 1 & 1 \end{pmatrix},\quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix},\quad B = \begin{pmatrix} 6 \\ 7 \\ 12 \end{pmatrix}$

   3. Find $|A|$ Compute the determinant:

   $|A| = 1 \cdot \begin{vmatrix} 0 & 2 \\ 1 & 1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 2 \\ 3 & 1 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 0 \\ 3 & 1 \end{vmatrix}$

   $= 1 \cdot (0\cdot1 - 2\cdot1) - 1 \cdot (1\cdot1 - 2\cdot3) + 1 \cdot (1\cdot1 - 0\cdot3)$

   $= 1 \cdot (-2) - 1 \cdot (1 - 6) + 1 \cdot (1) = -2 - (-5) + 1 = -2 + 5 + 1 = 4$

   So $|A| = 4$.

   4. Use the property of adjugate determinant For any square matrix of order $n$,

   $|\operatorname{adj} A| = |A|^{n-1}$

   Here $n = 3$, so:

   $|\operatorname{adj} A| = |A|^{3-1} = |A|^2 = 4^2 = 16$

   Tip

   A common mistake is to think $|\operatorname{adj} A| = |A|$ — but the correct formula is $|A|^{n-1}$. For a $3\times3$ matrix, that’s $|A|^2$.

   Watch out

   If you accidentally compute the adjugate explicitly, you’d still get the same result, but it’s far more work. Always use the determinant property for speed.

   ✓Final answer

   The correct option is (D).

   ANSWER: D

8. COMEDK 2024Set 2024-A1 markMCQ
   Q.

   [!FORMULA] $\text { If A }(\operatorname{adj} A)=5 I \text {, where I is the identity matrix of order } 3 \text {, then }|\operatorname{adj} A|=$

   (A) 5 (B) 125 (C) 25 (D) 10

   ›Reveal solutionSolution

   The key idea is that for a square matrix, $A(\operatorname{adj} A) = |A| I$. Given $A(\operatorname{adj} A) = 5I$ and the order is 3, we find $|A| = 5$ and then use $|\operatorname{adj} A| = |A|^{n-1} = 5^{2} = 25$. The correct option is (C).

   We start with a fundamental property of adjugates: for any square matrix $A$ of order $n$,

   $A (\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I.$

   This is the definitional relationship — the adjugate is the transpose of the cofactor matrix, and multiplying by $A$ yields a diagonal matrix where every diagonal entry is the determinant $|A|$.

   Here we are told $A (\operatorname{adj} A) = 5I$ and the order is 3. That means $n = 3$. Comparing with the property, we immediately see that $|A| I = 5I$, so $|A| = 5$.

   Now we need $|\operatorname{adj} A|$. There is a well-known formula: for an $n \times n$ matrix,

   $|\operatorname{adj} A| = |A|^{n-1}.$

   Why? Because from $A (\operatorname{adj} A) = |A| I$, take determinants of both sides:

   $|A| \cdot |\operatorname{adj} A| = |A|^n.$

   If $|A| \neq 0$ (which it is, since 5 ≠ 0), divide both sides by $|A|$ to get $|\operatorname{adj} A| = |A|^{n-1}$.

   Applying this with $n = 3$ and $|A| = 5$:

   $|\operatorname{adj} A| = 5^{3-1} = 5^2 = 25.$

   Watch out

   A common mistake is to think $|\operatorname{adj} A| = |A|$ or to forget the exponent depends on the order. Here, if you mistakenly used $n=2$ or just took $|A|$ itself, you'd pick 5 (option A) or 125 (option B). Always check the order.

   Tip

   Notice that the problem never gives $A$ itself — only the product $A(\operatorname{adj} A)$. That’s enough because the property directly links that product to $|A|$. This is a classic shortcut: you don’t need to construct the matrix.

   Thus, step by step:

   1. Recognize $A(\operatorname{adj} A) = |A| I$ for any square matrix.
   2. Given $A(\operatorname{adj} A) = 5I$ and order 3, equate: $|A| I = 5I \Rightarrow |A| = 5$.
   3. Use the determinant-of-adjugate formula: $|\operatorname{adj} A| = |A|^{n-1}$.
   4. Substitute $n=3$, $|A|=5$: $|\operatorname{adj} A| = 5^{2} = 25$.

   ✓Final answer

   The correct option is (C).

   ANSWER: C

9. COMEDK 2024Set 2024-E1 markMCQ
   Q.

   [!FORMULA] $\text { If } A=\left[\begin{array}{cc} 5 a & -b \\ 3 & 2 \end{array}\right] \text { and } A \operatorname{adj} A=A A^t \text {, then } 5 a+b \text { is equal to }$

   (A) 5 (B) $-$1 (C) 4 (D) 13

   ›Reveal solutionSolution

   The key idea is to use the property $A \operatorname{adj} A = |A| I$ and equate it to $A A^t$, then compare entries to solve for $a$ and $b$. The result is $5a + b = 5$.

   We are given

   $A = \begin{pmatrix} 5a & -b \\ 3 & 2 \end{pmatrix}$

   and the condition

   $A \operatorname{adj} A = A A^t.$

   We need $5a + b$.

   ---

   Concept and Intuition

   For any square matrix $A$, a fundamental identity is

   $A \operatorname{adj} A = |A| I,$

   where $|A|$ is the determinant and $I$ is the identity matrix. This is often faster than computing the adjugate explicitly. The right-hand side $A A^t$ is a product we can compute directly. So we set

   $|A| I = A A^t,$

   which gives us a system of equations by comparing entries.

   ---

   Step-by-step solution

   1. Compute $|A|$

   $|A| = (5a)(2) - (-b)(3) = 10a + 3b.$

   2. Write the left-hand side

   $A \operatorname{adj} A = |A| I = (10a + 3b) \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 10a+3b & 0 \\ 0 & 10a+3b \end{pmatrix}.$

   3. Compute $A A^t$ First, $A^t = \begin{pmatrix} 5a & 3 \\ -b & 2 \end{pmatrix}$. Then

   $A A^t = \begin{pmatrix} 5a & -b \\ 3 & 2 \end{pmatrix} \begin{pmatrix} 5a & 3 \\ -b & 2 \end{pmatrix} = \begin{pmatrix} (5a)(5a) + (-b)(-b) & (5a)(3) + (-b)(2) \\ (3)(5a) + (2)(-b) & (3)(3) + (2)(2) \end{pmatrix}.$

   Simplify each entry:

   • Top-left: $25a^2 + b^2$
   • Top-right: $15a - 2b$
   • Bottom-left: $15a - 2b$ (same)
   • Bottom-right: $9 + 4 = 13$

   So

   $A A^t = \begin{pmatrix} 25a^2 + b^2 & 15a - 2b \\ 15a - 2b & 13 \end{pmatrix}.$

   4. Equate the two matrices From $A \operatorname{adj} A = A A^t$ we have:

   $\begin{pmatrix} 10a+3b & 0 \\ 0 & 10a+3b \end{pmatrix} = \begin{pmatrix} 25a^2 + b^2 & 15a - 2b \\ 15a - 2b & 13 \end{pmatrix}.$

   This gives three equations (the off-diagonals give the same condition):

   • (1) $10a + 3b = 25a^2 + b^2$
   • (2) $0 = 15a - 2b$ → $15a = 2b$ → $b = \frac{15}{2}a$
   • (3) $10a + 3b = 13$

   5. Solve the system From (2): $b = \frac{15}{2}a$. Substitute into (3):

   $10a + 3\left(\frac{15}{2}a\right) = 13 \quad\Rightarrow\quad 10a + \frac{45}{2}a = 13.$

   Multiply by 2: $20a + 45a = 26$ → $65a = 26$ → $a = \frac{26}{65} = \frac{2}{5}$.

   Then $b = \frac{15}{2} \cdot \frac{2}{5} = 3$.

   Check equation (1):

   Left: $10a+3b = 10\cdot\frac{2}{5} + 3\cdot3 = 4 + 9 = 13$.

   Right: $25a^2 + b^2 = 25\cdot\frac{4}{25} + 9 = 4 + 9 = 13$. ✓

   6. Compute $5a + b$

   $5a + b = 5\cdot\frac{2}{5} + 3 = 2 + 3 = 5.$

   Tip

   The off-diagonal entry being zero is the key: it forces a direct relation between $a$ and $b$, making the system easy to solve.

   Watch out

   A common mistake is to forget that $A \operatorname{adj} A = |A| I$ only holds for square matrices, and to try computing the adjugate explicitly — that works but is slower. Using the determinant identity is much cleaner.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

10. COMEDK 2024Set 2024-M1 markMCQ
    Q.

    [!FORMULA] $\text { If } P=\left[\begin{array}{lll} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{array}\right] \text { is the adjoint of a } 3 \times 3 \text { matrix } A \text { and }|A|=4 \text { then } \alpha \text { is equal to }$

    (A) 11 (B) 4 (C) 0 (D) 5

    ›Reveal solutionSolution

    The key idea is that for a $3\times3$ matrix $A$, $\operatorname{adj}(A) = |A| A^{-1}$, so $P$ must satisfy $P = 4 A^{-1}$. Taking determinants gives $|P| = 4^2 = 16$, which yields an equation for $\alpha$. Solving gives $\alpha = 11$, so the correct option is (A).

    We are told that $P$ is the adjoint of $A$, and $|A| = 4$. For any invertible square matrix $A$, the fundamental relation is

    $A \cdot \operatorname{adj}(A) = |A| I.$

    Thus $\operatorname{adj}(A) = |A| A^{-1}$. Here $P = \operatorname{adj}(A)$, so

    $P = 4 A^{-1}.$

    Taking determinants on both sides:

    $|P| = |4 A^{-1}| = 4^3 |A^{-1}| = 64 \cdot \frac{1}{|A|} = \frac{64}{4} = 16.$

    So we must have $|P| = 16$. This gives an equation for $\alpha$.

    Now compute $|P|$:

    $P = \begin{pmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{pmatrix}.$

    1. Expand along the first row:

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

    2. Compute the $2\times2$ determinants:

    $\begin{vmatrix} 3 & 3 \\ 4 & 4 \end{vmatrix} = 3\cdot4 - 3\cdot4 = 0,$

    $\begin{vmatrix} 1 & 3 \\ 2 & 4 \end{vmatrix} = 1\cdot4 - 3\cdot2 = 4 - 6 = -2.$

    3. Substitute:

    $|P| = 1\cdot 0 - \alpha \cdot (-2) + 3 \cdot (-2) = 0 + 2\alpha - 6 = 2\alpha - 6.$

    Set this equal to $16$:

    $2\alpha - 6 = 16 \quad\Rightarrow\quad 2\alpha = 22 \quad\Rightarrow\quad \alpha = 11.$

    Watch out

    A common mistake is to forget that $|kA| = k^n |A|$ for an $n\times n$ matrix. Here $n=3$, so $|4A^{-1}| = 4^3 |A^{-1}|$, not $4|A^{-1}|$. Also, note that $|A^{-1}| = 1/|A|$, not $|A|$.

    Tip

    You can also use the property $|\operatorname{adj}(A)| = |A|^{n-1}$ directly. For $n=3$, $|P| = |A|^{2} = 4^2 = 16$, which is faster. Then compute $|P|$ as above.

    ✓Final answer

    The correct option is (A).

    ANSWER: A

11. KCET 2023Set A-21 markMCQ
    Q.If $A = \begin{bmatrix} 2-k & 2 \\ 1 & 3-k \end{bmatrix}$ is singular matrix, then the value of $5k - k^2$ is equal to (A) $-6$ (B) $-4$ (C) $6$ (D) $4$

    ›Reveal solutionSolution

    "Singular" means the determinant is zero; expand it, and the resulting quadratic is $k^2-5k+4=0$ — read $5k-k^2$ straight off it without ever solving for $k$.

    Step 1 — Condition for a singular matrix.

    A square matrix is singular exactly when it has no inverse, i.e. when

    $\det A=0.$

    Step 2 — Expand the determinant.

    $\det A=\begin{vmatrix}2-k & 2\\ 1 & 3-k\end{vmatrix}=(2-k)(3-k)-(2)(1)$

    $=(6-2k-3k+k^2)-2=k^2-5k+4$

    Step 3 — Set it to zero.

    $k^2-5k+4=0\quad\Longrightarrow\quad k^2-5k=-4$

    Step 4 — Read off the required expression.

    $5k-k^2=-(k^2-5k)=-(-4)=4$

    Cross-check by solving: $k^2-5k+4=0\Rightarrow(k-1)(k-4)=0\Rightarrow k=1$ or $k=4$.

    • $k=1$: $5(1)-1^2=4$ ✓
    • $k=4$: $5(4)-4^2=20-16=4$ ✓

    Both roots give the same value — the expression is well-defined.

    ✓Final answer

    The correct option is (D) — $4$.

    ANSWER: D

12. COMEDK 2023Set 2023-M1 markMCQ
    Q.If $A$ is a matrix of order $4 \operatorname{such}$ that $A(\operatorname{adj} A)=10 \mathrm{~I}$, then $|\operatorname{adj} A|$ is equal to (A) 10 (B) 100 (C) 1000 (D) 10000

    ›Reveal solutionSolution

    Since $A(\text{adj }A)=|A|I=10I$, we get $|A|=10$; then $|\text{adj }A|=|A|^{n-1}=10^{3}=1000$ for order $4$.

    Property: $A(\text{adj }A)=|A|\,I$. Comparing with $A(\text{adj }A)=10I$ gives $|A|=10$.

    For an $n\times n$ matrix, $|\text{adj }A|=|A|^{\,n-1}$. Here $n=4$, so

    $|\text{adj }A|=10^{4-1}=10^3=1000.$

    ✓Final answer

    The correct option is (C) — 1000