Q1811: Given that $P=\begin{bmatrix}2&-1\\3&4\end{bmatrix}$, $Q=\be...

Class CBSE Class 12 Mathematics Matrices and Determinants Q #1811

COMPETENCY BASED

APPLY

5 Marks 2026 AISSCE(Board Exam) LA

Given that $P=\begin{bmatrix}2&-1\\3&4\end{bmatrix}$, $Q=\begin{bmatrix}5&2\\7&4\end{bmatrix}$ and $R=\begin{bmatrix}2&5\\3&8\end{bmatrix}$, find a matrix S such that PQ - RS is a null matrix.

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AI Tutor Explanation

Detailed Solution

Step 1: Understand the Equation

We are given the equation $PQ - RS = O$, where $O$ is the null matrix. This implies $PQ = RS$. To find $S$, we need to isolate it by multiplying both sides by the inverse of $R$ from the left: $R^{-1}PQ = S$.

Step 2: Calculate the Product PQ

Multiply matrices $P$ and $Q$: $$PQ = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix} = \begin{bmatrix} (2)(5)+(-1)(7) & (2)(2)+(-1)(4) \\ (3)(5)+(4)(7) & (3)(2)+(4)(4) \end{bmatrix} = \begin{bmatrix} 3 & 0 \\ 43 & 22 \end{bmatrix}$$

Step 3: Find the Inverse of R

For $R = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix}$, the determinant $|R| = (2)(8) - (5)(3) = 16 - 15 = 1$. The inverse $R^{-1} = \frac{1}{|R|} \text{adj}(R) = \frac{1}{1} \begin{bmatrix} 8 & -5 \\ -3 & 2 \end{bmatrix} = \begin{bmatrix} 8 & -5 \\ -3 & 2 \end{bmatrix}$.

Step 4: Solve for S

Now compute $S = R^{-1}(PQ)$: $$S = \begin{bmatrix} 8 & -5 \\ -3 & 2 \end{bmatrix} \begin{bmatrix} 3 & 0 \\ 43 & 22 \end{bmatrix} = \begin{bmatrix} (8)(3)+(-5)(43) & (8)(0)+(-5)(22) \\ (-3)(3)+(2)(43) & (-3)(0)+(2)(22) \end{bmatrix}$$ $$S = \begin{bmatrix} 24 - 215 & 0 - 110 \\ -9 + 86 & 0 + 44 \end{bmatrix} = \begin{bmatrix} -191 & -110 \\ 77 & 44 \end{bmatrix}$$

Final Answer: S = \begin{bmatrix} -191 & -110 \\ 77 & 44 \end{bmatrix}

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student must utilize the properties of matrix algebra and inverse matrices to solve for an unknown variable matrix.

Knowledge Dimension: PROCEDURAL

Justification: The problem requires a sequence of specific mathematical operations (multiplication, finding determinants, finding adjoints, and matrix inversion) to reach the solution.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the student's ability to manipulate matrix equations beyond simple arithmetic, aligning with the NEP 2020 focus on conceptual application.