Q1305: If $A=[\begin{bmatrix}-1&a&2\\ 1&2&x\\ 3&1&1\end{bmatrix}]$ ...

Class CBSE Class 12 Mathematics Matrices and Determinants Q #1305

KNOWLEDGE BASED

REMEMBER

5 Marks 2024 AISSCE(Board Exam) LA

If $A=[\begin{bmatrix}-1&a&2\\ 1&2&x\\ 3&1&1\end{bmatrix}]$ and $A^{-1}=[\begin{bmatrix}1&-1&1\\ -8&7&-5\\ b&y&3\end{bmatrix}],$ find the value of $(a+x)-(b+y)$.

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AI Tutor Explanation

Detailed Solution

Step 1: Understanding the Problem

We are given a matrix $A$ and its inverse $A^{-1}$. We need to find the values of $a, x, b,$ and $y$ and then compute $(a+x)-(b+y)$. We know that $A \cdot A^{-1} = I$, where $I$ is the identity matrix.

Step 2: Multiplying $A$ and $A^{-1}$

We multiply the given matrix $A$ and $A^{-1}$: $$A \cdot A^{-1} = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Step 3: Calculating the elements of the resulting matrix

Let's calculate the elements of the resulting matrix by performing the matrix multiplication: \r\nElement (1,1): $(-1)(1) + a(-8) + 2(b) = 1 \implies -1 - 8a + 2b = 1 \implies -8a + 2b = 2 \implies -4a + b = 1$ (1)\r\nElement (1,2): $(-1)(-1) + a(7) + 2(y) = 0 \implies 1 + 7a + 2y = 0 \implies 7a + 2y = -1$ (2)\r\nElement (1,3): $(-1)(1) + a(-5) + 2(3) = 0 \implies -1 - 5a + 6 = 0 \implies -5a = -5 \implies a = 1$\r\nElement (2,1): $(1)(1) + 2(-8) + x(b) = 0 \implies 1 - 16 + bx = 0 \implies bx = 15$ (3)\r\nElement (2,2): $(1)(-1) + 2(7) + x(y) = 1 \implies -1 + 14 + xy = 1 \implies xy = -12$ (4)\r\nElement (2,3): $(1)(1) + 2(-5) + x(3) = 0 \implies 1 - 10 + 3x = 0 \implies 3x = 9 \implies x = 3$\r\nElement (3,1): $(3)(1) + 1(-8) + 1(b) = 0 \implies 3 - 8 + b = 0 \implies b = 5$\r\nElement (3,2): $(3)(-1) + 1(7) + 1(y) = 0 \implies -3 + 7 + y = 0 \implies y = -4$\r\nElement (3,3): $(3)(1) + 1(-5) + 1(3) = 1 \implies 3 - 5 + 3 = 1 \implies 1 = 1$ (This is consistent)\r\n

Step 4: Solving for the unknowns

From Element (1,3), we found $a = 1$. \r\nFrom Element (2,3), we found $x = 3$. \r\nFrom Element (3,1), we found $b = 5$. \r\nFrom Element (3,2), we found $y = -4$. \r\nLet's verify these values with the other equations:\r\nFrom (1): $-4a + b = 1 \implies -4(1) + 5 = 1 \implies 1 = 1$ (Correct)\r\nFrom (2): $7a + 2y = -1 \implies 7(1) + 2(-4) = -1 \implies 7 - 8 = -1 \implies -1 = -1$ (Correct)\r\nFrom (3): $bx = 15 \implies 5(3) = 15 \implies 15 = 15$ (Correct)\r\nFrom (4): $xy = -12 \implies 3(-4) = -12 \implies -12 = -12$ (Correct)

Step 5: Calculating $(a+x)-(b+y)$

Now we can calculate $(a+x)-(b+y)$: \r\n$(a+x)-(b+y) = (1+3) - (5+(-4)) = 4 - (5-4) = 4 - 1 = 3$

Final Answer: 3

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because the student needs to recall the property of inverse matrices ($A \cdot A^{-1} = I$) and apply matrix multiplication.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply the procedure of matrix multiplication and solving linear equations to find the unknown variables.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of matrix algebra and inverse matrices, which are core concepts in the syllabus.