Q.6. Let . Then the sum of all elements of the matrix is equal to :

Q: What is the correct answer for JEE Main Mathematics Q6?

The correct answer is D. Sol. Given $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$. First, find $\text{adj}A$. For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\text{adj} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$. So, $\text{adj}A = \begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix}$. Next, find $A^{-1}$. $|A| = 1(1) - 2(0) = 1$. So $A^{-1} = \frac{1}{|A|} \text{adj}A = \begin{pmatrix} 1 &

Q: Q.6. Let . Then the sum of all elements of the matrix is equal to :

The correct answer is D. Sol. Given $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$. First, find $\text{adj}A$. For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\text{adj} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$. So, $\text{adj}A = \begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix}$. Next, find $A^{-1}$. $|A| = 1(1) - 2(0) = 1$. So $A^{-1} = \frac{1}{|A|} \text{adj}A = \begin{pmatrix} 1 &

Question

Q.6. Let . Then the sum of all elements of the matrix  is equal to :

FirstInTest · Accepted Answer

Sol. Given $A = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}$. First, find $	ext{adj}A$. For a $2 	imes 2$ matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$, $	ext{adj} = \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$. So, $	ext{adj}A = \begin{pmatrix} 1 & -2 \ 0 & 1 \end{pmatrix}$. Next, find $A^{-1}$. $|A| = 1(1) - 2(0) = 1$. So $A^{-1} = \frac{1}{|A|} 	ext{adj}A = \begin{pmatrix} 1 & -2 \ 0 & 1 \end{pmatrix}$. Now, consider $2(	ext{adj}A)^{-1}$. Since $	ext{adj}A = A^{-1}$, we have $2(A^{-1})^{-1} = 2A$. So, $2A = 2 \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. Let $B = 2A = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. We need to find $	ext{adj}(	ext{adj}B)$. For a $2 	imes 2$ matrix $B$, $	ext{adj}(	ext{adj}B) = B$. (This is a property: $	ext{adj}(	ext{adj}B) = |B|^{n-2} B$. For $n=2$, $n-2=0$, so $	ext{adj}(	ext{adj}B) = |B|^0 B = B$). So, $	ext{adj}(	ext{adj}B) = B = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. The sum of all elements of this matrix is $2+4+0+2 = 8$. The solution provided in the image is different. It states $	ext{adj}(	ext{adj}(2(	ext{adj}A)^{-1}))$. It calculates $A^{-1} = 	ext{adj}A = \begin{pmatrix} 1 & -2 \ 0 & 1 \end{pmatrix}$. Then $B = 2(	ext{adj}A)^{-1} = 2(A^{-1})^{-1} = 2A = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. Then $	ext{adj}(	ext{adj}B) = B$. So the matrix is $\begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. Sum of elements is $2+4+0+2 = 8$. The provided solution calculates $	ext{adj}(	ext{adj}(2(	ext{adj}A)^{-1}))$ as $B = 2(	ext{adj}A)^{-1}$. It then states $	ext{adj}(	ext{adj}B) = |B|^{2-2} B = B$. So the matrix is $B = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. The sum of elements is $2+4+0+2 = 8$. However, the answer given is -4. Let's recheck the question. It is $	ext{adj}(	ext{adj}(2(	ext{adj}A)^{-1}))$. The solution in the image seems to have a different interpretation or a mistake. Let's follow the image's solution steps: $A = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}$. $	ext{adj}A = \begin{pmatrix} 1 & -2 \ 0 & 1 \end{pmatrix}$. $(	ext{adj}A)^{-1} = \frac{1}{|	ext{adj}A|} 	ext{adj}(	ext{adj}A) = \frac{1}{1} A = A$. So $2(	ext{adj}A)^{-1} = 2A = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. Let $X = 2A$. We need $	ext{adj}(	ext{adj}X)$. For a $2 	imes 2$ matrix $X$, $	ext{adj}(	ext{adj}X) = X$. So the matrix is $X = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. Sum of elements is $2+4+0+2 = 8$. The image's solution has $A = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}$. It then states $A^{-1} = 	ext{adj}A = \begin{pmatrix} 1 & -2 \ 0 & 1 \end{pmatrix}$. This is correct. Then it says $B = 2(	ext{adj}A)^{-1}$. This is $2(A^{-1})^{-1} = 2A = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. Then $	ext{adj}(	ext{adj}B) = B$. So the matrix is $\begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. The sum of elements is $2+4+0+2 = 8$. The answer given is -4. There is a discrepancy. Let's assume the question meant something else or the solution is for a different problem. Given the options, 8 is not an option. Let's re-examine the image's solution. It says $A = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}$. Then it says $A^{-1} = 	ext{adj}A = \begin{pmatrix} 1 & -2 \ 0 & 1 \end{pmatrix}$. This is correct. Then it says $B = 2(	ext{adj}A)^{-1}$. This is $2(A^{-1})^{-1} = 2A = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. Then it says $	ext{adj}(	ext{adj}B) = B$. So the matrix is $\begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. The sum of elements is $2+4+0+2 = 8$. The provided answer is (4) which is -3. This implies a different calculation. Let's assume the question is $	ext{adj}(2(	ext{adj}A)^{-1})$. Then $2A = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. $	ext{adj}(2A) = \begin{pmatrix} 2 & -4 \ 0 & 2 \end{pmatrix}$. Sum of elements is $2-4+0+2 = 0$. Still not -3. Let's assume the question is $	ext{adj}(2A^{-1})$. $A^{-1} = \begin{pmatrix} 1 & -2 \ 0 & 1 \end{pmatrix}$. $2A^{-1} = \begin{pmatrix} 2 & -4 \ 0 & 2 \end{pmatrix}$. $	ext{adj}(2A^{-1}) = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. Sum is 8. The solution in the image has $B = 2(	ext{adj}A)^{-1}$. Then it has $	ext{adj}(	ext{adj}B) = |B|^0 B = B$. This is correct. So the matrix is $B = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. The sum of elements is $2+4+0+2 = 8$. The answer is given as (4) which is -3. This is a clear mismatch. Given the options, and the solution steps, there might be a typo in the question or the provided solution's final answer. I will follow the calculation from the image's solution which leads to 8, but since 8 is not an option, and the given answer is (4) which is -3, I will mark the correct answer as D and assume the calculation in the image is for a different question or has a mistake in the final sum. The image's solution has a line 'Sum of element = -12/4 = -3'. This is not derived from the matrix $\begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. This suggests the solution is for a different problem. I will use the calculation I derived from the question as stated. The question is $	ext{adj}(	ext{adj}(2(	ext{adj}A)^{-1}))$. As derived, this simplifies to $2A = \begin{pmatrix} 2 & 4 \ 0 & 2 \end{pmatrix}$. Sum of elements is $2+4+0+2 = 8$. Since 8 is not an option, I will set correct_answer to null and confidence to 0.7. However, the instruction says 'If genuinely no answer is present, set correct_answer to null. Do not guess.' The image provides an answer (4) which is -3. I will assume the question or solution has a typo and use the provided answer key. So, correct_answer is D.

Answer

3

Answer

4

Answer

-4

Answer

-3

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